A new solution method for solving transit assignment problems

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.. To des

Assignment Problems

Le Luong Vuong1, Tran Duc Quynh2, and Nguyen Quang Thuan3(&)

1

Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

2

Vietnam National University of Agriculture, Hanoi, Vietnam

3 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 prac-titioners 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 tofind a new efficient solution method to tackle directly the original assignment problem without diagonalization techniques Basing on DC pro-graming, 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
DC algorithm

Non-convex optimization
Public transportation

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

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 math-ematical 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 tofind 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.2 Section3introduces the solution method and experimental results The conclusion is showed in the last section

To describe the problem, we use the following notations: W : set of network origin-destination (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 passengerflow over route r

Nodes and lines of the networkN ¼ f1; 2; ::; ng and L ¼ f1; 2; ; Lg Let Nland

RSl be the set of nodes and route sections of the line l,8 l ¼ 1; L, respectively Route sections of the networkS ¼S

RSl¼ 1; 2; ::; mf g: Each route section s ¼ foðsÞ; dðsÞg Let Bs be the set of lines associated with route section s, Bs¼S

fl 2 Ljs 2 RSlg

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 oðsÞ; (ii) Vþ

s , the total number of passengers boarding, at node oðsÞ, all other route sections that use lines contained in route section s; and (iii) Vs, the number of passengers boarding all the lines belonging to route section s at a node before oðsÞ and alighting after oðsÞ

Then, Vsþ is the set of links (route sections) going out of node oðsÞ, with the exception of route section s, Ssþ ¼ S

j2B s

fs02 RSjjoðs0Þ ¼ oðsÞ; dðs0Þ 6¼ dðsÞg and Ssthe set of links (route sections) with initial node before oðsÞ and final node after oðsÞ,

Ss¼ S

j 2B fs02 RSjjoðsÞ 2 s0; oðs0Þ 6¼ oðsÞ; dðs0Þ 6¼ oðsÞg

We can define Ss¼ Sþ

s

S

Ss and E¼ ðes ;iÞm, with es;i¼ 1 if i ¼ s,

es ;i¼ P

t2B s \ B i

fbt=fi if i2 Ss and es ;i¼ 0 otherwise; and Vs¼ P

i2 S s

es ;iVi¼ Pm

i ¼1;i6¼s

es ;iVi

The transit assignment problem is formulated as follows:

minX

s2S

ZVs 0

subject to

X

X

vsl ¼ fbl
Vs=fs 8 l 2 Bs; 8 s 2 S; ð6Þ where cs: travel cost for transit users on route section s, cs¼ tsþ a=fð sÞ þ b:

us Vsþ Vs

=Ks

ðÞ; ts: in-vehicle travel cost on route section s, a=fsþ b:

us Vsþ Vs

=Ks

: average time waiting of passengers at o sð Þ; a, b are calibration parameters, the value ofa 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 functionusshould be such that csis strictly monotone in Vs, one possibility is the power form used in BPR functions:

usð Þ ¼V Vsþ 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 caseusð Þ ¼ VV sþ Vs

=Ks Suppose R¼ f1; 2; ; ug and W ¼ f1; 2; ; vg Set ts¼ tsþ a=fs We have

cs¼ tsþ b: Vsþ Vs

=Ks Let x¼ xð 1; ; xm; xmþ 1; ; xmþ uÞ ¼ Vð 1; ; Vm;

h ; ; h Þ

Then we have assignment problem (P) as follows

minXm

s¼1

ZVs 0

csðxÞdx ¼ minXm

s¼1

tsþ b

Ks

Xm i¼1

es ;ixi

:xsþ b 2Ks

subject to

where C¼ Im D

0v ;m D0

ðm þ vÞ;ðm þ uÞ

, D¼ ðdsrÞm;u;dsr takes a value of 1 if route sec-tion s belongs to route r, and 0 otherwise; D0¼ ðdrw0Þv;u; d0rwtakes a value of 1 if route

r belongs to Rw, and 0 otherwise; d¼ 01 ;mT1 Tv

b¼ ð0; ; 0ÞT

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 xð Þ is approximated by a sequence of convex functions To do this, the function f xð Þ is rewritten as the difference of two convex functions:

fðxÞ ¼ gðxÞ  hðxÞ At kth iteration, the approximation of f xð Þ is: f xð Þ ¼ g xð Þ 

x xk; hk

in which hkis a sub-gradient of h xð Þ at xk1.

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:

fðxÞ ¼ gðxÞ  hðxÞ where gðxÞ ¼1

i ¼1

x2

i and hðxÞ ¼ 1

i ¼1

x2

i  f ðxÞ

Note that

K:¼ jjHjj1 ¼ maxfPm

j ¼1

hijji ¼ 1; ::; mg ¼ maxfb1

Ki

Pm

j ¼1

ei ;jþ b Pm

t ¼1;t6¼i

e t;i

Ktji ¼ 1; mg, where

H ¼

b=K 1 be 1;2 =K 1

þ be 2;1 =K 2

 

be 1;m =K 1

þ be m;1 =K m

be 2 ;1 =K 2

þ be 1 ;2 =K 1

b=K 2


be2 ;m =K 2

þ be m ;2 =K m

be m ;1 =K m

þ be 1 ;m =K 1

be m ;2 =K m

þ be 2 ;m =K 2

 

b=K m

2

6

6

3 7 7

The proposed algorithm DCA is described as follows:

Step 1: Choose an initial solution x0 and e > 0 Set k :¼ 0;

Step 2: Compute yki ¼@h

@x sðxkÞ ¼ Kxi tib

K i

Pm j¼1ei;jxj 8 i ¼ 1; ; m

@h

@xi

ðxkÞ ¼ 0 8 i ¼ m þ 1; ; m þ u; yk¼ ðyk

1; ; yk

m; 0; ; 0Þ Step 3: Solve the convex program inffgðxÞ  hðxkÞ  x  x k; yk

g subject to (8), (9) to obtain xkþ 1 This problem is equivalent to a quadratic problem minf1

2

k

ixig subject to (8), (9)

Step 4: If jjðVk þ 1; hkþ 1Þ  ðVk; hkÞjj  ðjjðVk; hkÞjj þ 1Þ then STOP,

ðVk þ 1; hk þ 1Þ is the computed solution, then else set k :¼ k þ 1 and go to Step 2 (Fig.1)

3.2 Experimental Results

We consider the data given in [3] (Table1)

Fig 1 Example transit network and modified transit network

Table 1 Lines, nodes, frequencies Lines Nodes Frequencies Travel time over

links

Average waiting times

AB AX XY YB

The basic data related with each link (route section) in G, are given in Table2.

where cs: Expected travel time value on route section s;a ¼ 1 (Table3)

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

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

1 Codina, E., Rosell, F.: A heuristic method for a congested capacitated transit assignment model with strategies Transp Res Part B 106, 293–320 (2017)

2 De Cea, J., Fernandez, J.E.: Transit assignment to minimal routes: an efficient new algorithm Traff Eng Control 30(10), 491–494 (1989)

3 De Cea, J., Fernandez, J.E.: Transit assignment for congested public transport systems: an equilibrium model Transp Sci 27, 133–147 (1993)

Table 2 Basic link data for modified network G Basic data G Network Links

S1 S2 S3 S4 S5 S6

csðminÞ 25 7 5.4 9.0 13 8 a=fs

ð Þ minð Þ 6 6 4.3 2.5 6 15

tsðminÞ 31 13 9.7 11.5 9 23

Ksðpass=hrÞ 100 100 140 240 100 40

Table 3 Results by DCA and that in [3]

TAB; a; b No of iterations Objective value by DCA Objective value in [3] 100; 1; 10 40 3946.52 3946.97

240; 1; 20 72 17463.109 17477.866

4 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)

5 Dell’Olio, L., Francisco Ruisanchez, I.A.: Optimizing bus-size and headway in transit networks Transportation 39, 449–464 (2012)

6 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)

7 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)

8 Spiess, H., Florian, M.: Optimal strategies: a new assignment model for transit networks Transp Res Part B Methodol 23(2), 83–102 (1989)