A New Solution Method for Solving Transit Assignment Problems Le Luong Vuong1, Tran Duc Quynh2, and Nguyen Quang Thuan3(&) Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam Vietnam National University of Agriculture, Hanoi, Vietnam International School (VNU-IS), Vietnam National University-Hanoi (VNU), Hanoi, Vietnam nguyenquangthuan@vnu.edu.vn Abstract Congested transit assignment problems are crucial sub problems in planning public transportation systems These problems are usually formulated in the form of non-convex optimization programs In this work, we investigate the model given by De Cea et al [3] that has been widely used by both practitioners and researchers For solving this model, to the best of our knowledge, one must use a diagonalization technique in order to yield a symmetric assignment problem before applying a solution method Consequently, the quality of the obtained solution would be possibly affected The motivation of our work is to find a new efficient solution method to tackle directly the original assignment problem without diagonalization techniques Basing on DC programing, we introduce a new solution method The proposed algorithm is tested on the data given in [3] Comparing with the existing method, the experimental results show that our approach is promising Keywords: Transit assignment problems Á DC algorithm Non-convex optimization Á Public transportation Introduction Public transportation is a key factor of urban transportation that occupies an important place in economic and social development of a country Some problems can be listed such as transit route network design, bus scheduling, bus rapid transit systems, etc These problems have attracted attention from both practitioners and researchers In general, these problems are formulated into optimization problems and then are solved by certain solution methods The mathematical formulations are usually in the form of a bi-level optimization problem in which the lower level is a transit assignment problem [4–6] Finding efficient algorithms for the lower problem plays an important role in the schema of solving the original problems In case that congestion is not taken into account, assignment problems are possibly modeled as linear programs for which efficient solution algorithms have been implemented [2, 8] However, linear models are limited since congestion is often encountered in cities Some attempts have been made in the past to build transit assignment models that consider congestion [1, 3] © Springer Nature Switzerland AG 2019 H Fujita et al (Eds.): ICERA 2018, LNNS 63, pp 70–76, 2019 https://doi.org/10.1007/978-3-030-04792-4_11 A New Solution Method for Solving Transit Assignment Problems 71 In [3], the authors presented a congested model based on the concept of a “transit route” in 1993 Since then, this non-linear model is used in many different problems such as problems of locating bus stops and optimizing frequencies [4], problems of optimizing bus stop spacing [6], problems of optimizing bus size and headway [5] The model was solved by a diagonalization method [3] To the best of our knowledge, there does not exist another method to solve this model In this work, we introduce a new alternative solution method based on the mathematical technique in non-convex optimization, namely, DC programming and DCA This technique has been successfully applied to many non-convex optimization problems and showed the efficiency in particular for large-scale problems [7] To apply DCA, we have to find a suitable reformulation for the original problem Obviously, we first decompose the objective function as difference of two convex functions then propose a new schema of DC algorithm The algorithm is tested on the data given in [3] The experimentation shows that our result is a little better than the one given by the existing method The structure of the paper is organized as follows After the introduction section, we present the problem description in Sect Section introduces the solution method and experimental results The conclusion is showed in the last section Problem Description To describe the problem, we use the following notations: W: set of network origindestination (O-D) pairs; w: an element of set W; R: set of routes in G available to transit users; Rw : set of feasible routes associated with O-D pair w; r: an element of set R; As : set of all transit lines going from the origin node to the destination node of route section s; oðsÞ, dðsÞ: origin and destination node of route section s; fbl : frequency of line l; kl : practical capacity of line l; fs : total frequency on route section s; Ks practical capacity on route section s; T: set of O-D transit demands; Tw : transit demand between O-D pair w For the model internal variables, we will use: Vs : transit passenger flow on route section s; hr transit passenger flow over route r Nodes and lines of the network N ¼ f1; 2; ::; ng and L ¼ f1; 2; ; Lg Let Nl and RSl be the set of nodes and route S sections of the line l, l ¼ 1; L, respectively Route sections of the network S ¼ RSl ¼ f1; 2; ::; mg: Each route section S s ẳ fosị; dsịg Let Bs be the set of lines associated with route section s, Bs ¼ fl Ljs RSl g In general, the waiting time of a passenger boarding the route section s (at its origin node oðsÞ) will depend on: (i) Vs , the total number of passengers boarding the same route section, at node osị; (ii) Vsỵ , the total number of passengers boarding, at node s , the oðsÞ, all other route sections that use lines contained in route section s; and (iii) V number of passengers boarding all the lines belonging to route section s at a node before osị and alighting after osị Then, Vsỵ is the set of links (route sections) going out of node oðsÞ, with the S exception of route section s, Ssỵ ẳ fs RSj jos0 ị ẳ osị; ds0 ị 6ẳ dðsÞg and Ss the j2Bs set of links (route sections) with initial node before oðsÞ and final node after osị, S Ss ẳ fs RSj josị s0 ; os0 ị 6ẳ osị; ds0 ị 6ẳ osịg j2Bs 72 L L Vuong et al S We can dene Ss ẳ Ssỵ Ss and E ẳ es;i ịm , with es;i ¼ if i ¼ s, m P P P es;i ¼ fbt =fi if i Ss and es;i ¼ otherwise; and Vs ¼ es;i Vi ¼ es;i Vi t2Bs \ Bi i2 Ss i¼1;i6¼s The transit assignment problem is formulated as follows: XZ Vs s2S cs ðxÞdx ð1Þ subject to X hr ẳ Tw w W; 2ị dsr hr ¼ Vs s SÉ; ð3Þ r2Rw X r2R hr ! r R; ð4Þ Vs ! s S; 5ị vsl ẳ fbl Vs =fs l Bs ; s S; ð6Þ where cs : travel cost for transit users on route section s, cs ẳ ts ỵ a=fs ị ỵ b: us Vs ỵ Vs =Ks ị; ts : in-vehicle travel cost on route section s, a=fs þ b: us Vs þ Vs =Ks : average time waiting of passengers at oðsÞ; a, b are calibration parameters, the value of a depends on the distribution assumed for buses interarrival times (headways) and passenger arrival times, the value of b depends on the level congestion; the third term on the right (*) takes explicitly into account the effect of congestion on waiting time and the form of function us should be such that cs is strictly monotone in Vs , one possibility is the power form used in BPR functions: n us ðV ị ẳ Vs ỵ Vs =Ks ; vsl : passengers traveling on line l, over route section s (line section flow) The objective function (1) is the total time of all passengers in the network The constraints (2), (3), (4) and (5) ensure the equilibrium conditions (Wardropian conditions over the transit network) Constraint (6) is used to calculate the flow on the route section belonging to a line For simplicity we consider the following case us ðV Þ ẳ Vs ỵ Vs =Ks Suppose R ẳ f1; 2; ; ug and W ¼ f1; 2; ; vg Set ts ẳ ts ỵ a=fs We have Let x ¼ ðx1 ; ; xm ; xm ỵ ; ; xm ỵ u ị ẳ V1 ; ; Vm ; cs ẳ ts ỵ b: Vs ỵ Vs =Ks h1 ; ; hu Þ A New Solution Method for Solving Transit Assignment Problems 73 Then we have assignment problem (P) as follows m Z X Vs sẳ1 cs xịdx ẳ m X sẳ1 " # m bX b ts ỵ es;i xi :xs ỵ x Ks iẳ1 2Ks s 7ị subject to Im where C ¼ 0v;m ÀD D0 Cx ẳ d; 8ị Ax 9ị b; ! m ỵ vị;m ỵ uị , D ẳ dsr ịm;u ;dsr takes a value of if route sec0 tion s belongs to route r, and otherwise; D0 ẳ drw0 ịv;u ; drw takes a value of if route À ÁT r belongs to Rw , and otherwise; d ¼ 01;m T1 Tv 1;m ỵ v ; A ẳ Im ỵ u ; b ẳ 0; ; 0ịTm ỵ u Solution Method and Experimental Results 3.1 DC Algorithm (DCA) In this section, we develop a new solution method based on DC programming for solving Problem (P) The idea of algorithm is quite simple: The non-convex objective function f ð xÞ is approximated by a sequence of convex functions To this, the function f ð xÞ is rewritten as the difference of two convex functions: f xị ẳ gxị hxị At kth iteration, the approximation of f ð xÞ is: f ð xị ẳ g xị x xk ; hk in which hk is a sub-gradient of hð xÞ at xkÀ1 The decomposition of the objective function is a crucial step to determine the efficiency of the algorithm The delicacy of the decomposition is in the following: m m P P 2 f xị ẳ gxị hxị where gxị ẳ K xi and hxị ẳ K xi f xị Note that K :ẳ jjHjj1 ẳ maxf m P j¼1 i¼1 hij ji ¼ 1; ::; mg ¼ maxfb K1i m P iẳ1 ei;j ỵ b jẳ1 m P t¼1;t6¼i et;i Kt ji ¼ 1; mg, where b=K be1;2 =K1 ỵ be2;1 =K2 be2;1 =K2 ỵ be1;2 =K1 b=K2 H¼6 À Á À Á À bem;1 =Km ỵ be1;m =K1 bem;2 =Km þ be2;m =K2 À ÁÁÁ ÁÁÁ ÁÁÁ À be1;m =K1 ỵ bem;1 =Km be2;m =K2 ỵ bem;2 =Km 7 b=Km 74 L L Vuong et al The proposed algorithm DCA is described as follows: Step 1: Choose an initial solution x0 and e > Set k :¼ 0; m P @h xk ị ẳ Kxi ti Kbi ei;j xj i ¼ 1; ; m Step 2: Compute yki ¼ @x s j¼1 @h k x ị ẳ i ẳ m ỵ 1; ; m ỵ u; yk ẳ yk1 ; ; ykm ; 0; ; 0Þ @xi Step 3: Solve the convex program inffgðxÞ À hðxk Þ À x À xk ; yk g subject to (8), (9) to obtain xk ỵ This problem is equivalent to a quadratic problem m m P P minf12 K x2i À yki xi g subject to (8), (9) i¼1 i¼1 Step 4: If jjðV k ỵ ; hk ỵ ị V k ; hk ịjj jjV k ; hk ịjj ỵ 1ị then STOP, V k ỵ ; hk ỵ Þ is the computed solution, then else set k :ẳ k ỵ and go to Step (Fig 1) Fig Example transit network and modified transit network 3.2 Experimental Results We consider the data given in [3] (Table 1) Table Lines, nodes, frequencies Lines Nodes Frequencies Travel time over links AB AX XY YB A, B 10 25 A, X, Y 10 X, Y, B 4 Y, B 20 10 Average waiting times L1 L2 L3 L4 6 15 A New Solution Method for Solving Transit Assignment Problems 75 The basic data related with each link (route section) in G, are given in Table Table Basic link data for modified network G Basic data G Network Links S1 S2 S3 S4 cs ðminÞ 25 5.4 9.0 ða=fs Þ ðminÞ 6 4.3 2.5 ts ðminÞ 31 13 9.7 11.5 Ks ðpass=hr Þ 100 100 140 240 S5 13 100 S6 15 23 40 where cs : Expected travel time value on route section s; a ¼ (Table 3) Table Results by DCA and that in [3] No of iterations Objective value by DCA Objective value in [3] TAB ; a; b 100; 1; 10 40 3946.52 3946.97 240; 1; 20 72 17463.109 17477.866 The algorithm is implemented in C The computing time is quite fast It gives the solution in seconds Comparing with the objective value obtained by the algorithm in [3], the result of DCA is better Conclusion In this paper, we have investigated a congested transit assignment problem The mathematical formulation is transformed to a DC programming problem and then we proposed a new algorithm based on DC schema to solve it The computation shows that our approach provides an alternative method and it is promising In future work, we plan to apply the algorithm to large scale problems and integrate it into bilevel programs References Codina, E., Rosell, F.: A heuristic method for a congested capacitated transit assignment model with strategies Transp Res Part B 106, 293–320 (2017) De Cea, J., Fernandez, J.E.: Transit assignment to minimal routes: an efficient new algorithm Traff Eng Control 30(10), 491–494 (1989) De Cea, J., Fernandez, J.E.: Transit assignment for congested public transport systems: an equilibrium model Transp Sci 27, 133–147 (1993) 76 L L Vuong et al Dell’Olio, L., Ibeas, A., y Moura, J.L.: A bi-level mathematical programming model to locate bus stops and optimize frequencies Transp Res Rec J Transp Res Board 1971, 23–31 (2006) Dell’Olio, L., Francisco Ruisanchez, I.A.: Optimizing bus-size and headway in transit networks Transportation 39, 449–464 (2012) Ibeas, A., dell’Olio, L., Alonso, B., Sainz, O.: Optimizing bus stop spacing in urban areas Transp Res Part E 46(3), 446–458 (2010) Le Thi, H.A., Pham Dinh, T.: The DC (difference of convex functions) Programming and DCA revisited with DC models of real wourd nonconvex optimization problems Annal Oper Res 133, 23–46 (2005) Spiess, H., Florian, M.: Optimal strategies: a new assignment model for transit networks Transp Res Part B Methodol 23(2), 83–102 (1989) ... mathematical formulation is transformed to a DC programming problem and then we proposed a new algorithm based on DC schema to solve it The computation shows that our approach provides an alternative method. .. congested capacitated transit assignment model with strategies Transp Res Part B 106, 293–320 (2017) De Cea, J., Fernandez, J.E.: Transit assignment to minimal routes: an efficient new algorithm Traff... of passengers at oðsÞ; a, b are calibration parameters, the value of a depends on the distribution assumed for buses interarrival times (headways) and passenger arrival times, the value of b depends