META-HEURISTIC ALGORITHMS FOR A TRANSIT ROUTE DESIGN

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1 Advanced OR and AI Methods in Transportation META-HEURISTIC ALGORITHMS FOR A TRANSIT ROUTE DESIGN Jongha HAN 1, Seungjae LEE 2, Jonghyung KIM 3 Absact. Since a Bus Transit Route Networ (BTRN) design problem leads to have multiple solutions in its nature, some meta-heuristic algorithms such as simulated annealing, genetic and tabu search algorithms have been developed in order to find a global optimum. Suggested approach for BTRN has been compared with the existing benchmar results. We have found that our solution is better than the other ones in some networ parameters. 1. Inoduction A Bus Transit Route Networ (BTRN) can be designed in terms of its set of lines and frequencies using meta-heuristic algorithms such as simulated annealing, genetic and tabu search algorithms. Since this problem is too complex, the existing studies have redefined this problem as a sequence of sub-problems: (1) Candidate Route Generation Part (CRGP), (2) Route Analysis/Evaluation Part (RAEP), (3) Route/Networ Improvement Part (RNIP) (see, Baaj and Mahmassani,1990; Shih and Mahmassani,1995, Weifan, 2004 among others). In this study, we improve the existing bus networ design procedure as the following three parts. The first, the CRGP includes candidate route set as the imal flow path as well as an existing shortest path. The second, Meta-heuristic algorithms developed guarantee that a near optimal solution can be obtained in non-convex bus ansit networ design problem using bi-level program based on iterative optimization process. The third, the RAEP uses ansit assignment considering ansfers between O-D pairs depending on the configuration of ansit networs and expected avel times after boarding as well as expected waiting time required for a route choice procedure. 1 Incheon Development Institute, San64-1 Shimgodong Seogu Incheon Korea, [email protected] 2 University of Seoul, 90 Cheonnongdong Dongdaemungu Seoul Korea, [email protected] 3 Incheon Development Institute, San64-1 Shimgodong Seogu Incheon Korea, [email protected]

2 Meta-heuristic algorithms for a ansit route design Model Formulation Since the BTRN problem is a mixed combinatorial problem, it is usually NP-hard. It is possible to ansform the original problem into bi-level problem classified as upper and lower level problem. Namely, the objective function of the upper level problem is to imize the sum of user cost, operator and demand costs for the bus ansit networ. On the other hand, the lower level problem is the bus ansit route choice and ip assignment assug that ansit avelers always attempt to complete his/her ip with an intention to avoid ansfers, and have the least total avel time as the following formulation:. [Upper level] Min. Z ( s( R, f ), d( s) ) [Lower level] d = DR, TR = c 1 i, + c d ( s) c 3 i, ( s) + r d r DR d t M r + O ( s) + r d i, r DR DR, TR d i, TR t d ( s) + c d i, TR M f t 2 = 1 r r ( (, (, ))) P c t s R f d = 0 and d ( (, (, ) P c t s R f ) d = 0 subject to f f f R L f L t W R R D D D R M R Notation: N = Nodes(Bus stop); r R = Routes; R = Transfer paths that use more than one route from R; R = Maximum allowed number of routes for the route networ; N = Number of nodes in the networ; D = Maximum length of any route in the ansit networ; D = Minimum length of any route in the ansit networ; f = Maximum frequency required for any route; f = Minimum frequency required for any route; L = Maximum load factor for any route; W = Maximum bus fleet size available for operations on the route networ; M = The number of routes of the current proposed bus ansit networ solution; D = The overall length of route r

3 688 J. Han et al r d = The bus avel demand between nodes i and j on route r d = The bus avel demand between nodes i and j along ansfer path TR = The set of direct routes used to serve the demand between nodes i and j. DR = The set of ansfer paths used to serve the demand between nodes i and j. r t = The total avel time between nodes i and j on route t = The total avel time between nodes i and j along ansfer path f = The bus frequency operating on route t = The round ip time of route r d (s) = The bus avel demand between nodes i and j on networ designed by s c (s) = The bus avel cost between nodes i and j on networ designed by s O (s) = The bus operator cost between nodes i and j on networ designed by s S ( R, f ) = The bus networ design variable composed of route set(r) and its frequency(f) d (s) = The bus avel demand on networ designed by s r c 1, c2, c3 = Weights reflecting the relative importance of three components including the user costs, operator costs and unsatisfied demand costs respectively r 3. Solution Algorithm procedure 3.1. Feasible Solution Space Existing research is focused on optimal bus ansit routes design based on shortest path. An optimal route may not be the shortest path but one that may tae longer to avel but serves more ansit users. Therefore, adeoffs between avel time and service coverage must be considered in optimizing a ansit networ. Shortest path(=minimum Time): MT Maximal flow path(=maximum Demand): MD Maximal flow path per Minimum Time: MDMT Maximal flow path per Minimum Route Length: MDML Maximal flow path per Minimum Cost: MDMC 3.2. Upper level: Meta-heuristic for bus ansit route optimal design A meta-heuristic algorithm: (i) conol the execution of a simpler heuristic method; and (ii) unlie a one-pass heuristic, does not automatically terate once a locally optimal solution is found. Thus, explicit teration condition must be set for meta-heuristic procedure that goes beyond feasibility or local optimality. We compared a solution searching efficiency (solution convergence and computing time) about each of three meta-heuristic methods:

4 Meta-heuristic algorithms for a ansit route design 689 genetic algorithm(ga), simulated annealing(sa), and tabu search(ts) Lower level: Bus ansit route choice and assignment The Lower level is an analytical tool that can evaluate and analyze a bus ansit networ and the route frequencies. To accomplish these tass, the lower level program employs an iterative procedure that sees to achieve internal consistency of route frequencies. Furthermore, it uses ansit assignment considering ansfers between O-D pairs depending on the configuration of ansit networs and expected avel times after boarding as well as expected waiting time required for a route choice procedure. START Feasible Solution Space Candidate Route Set Generation Process (1) (1) Upper Level Program Bus Transit Route Networ Design Based on Lower level program results Using a Optimal Solution Search Algorithm Networ & Frequency (2) Yes Lower Level Program Transit route choice and Transit ip assignment STOP Output the optimal ansit route set Newor Demand Maximum Number of Route? No Yes (2) Figure 1. BTRN solution procedure 4. Numerical Experimental: Mandl s ansit networ This networ has been utilized by Baaj and Mahmassani, Shih and Mahmassani, Lee, and Weifan as benchmar problem to compare their results with Mandl s solutions. In this section, we analyze which solution search algorithm is so fast converged in solution space. And we compared with optimal bus ansit route sets.

5 690 J. Han et al Figure 2. Mandl s ansit networ 4.1. Solution search algorithm s efficiency comparison A meta-heuristic algorithm has parameters that can decide on solution search convergence ability within a very large range. Several discrete values are chosen for each continuous parameter in all algorithms and the optimal parameter set is decided. Based on this optimal parameters, we compared with solution search efficiency: computing time and solution convergence ability. Computing time( sec/1000) 40,000 35,000 30,000 25,000 20,000 15,000 10,000 5,000 GA SA TS Iteration Objective function value 350, , , , , ,000 50,000 0 TS SA GA iteration Figure 3. Solution convergence ability Figure 4. Computing time ability 4.2. Optimal bus ansit route set comparison Suggested approach for BTRN is compared with the existing benchmar research results. We found that our solution is better than other the ones in some networ parameters.

6 Meta-heuristic algorithms for a ansit route design 691 Mandl Baaj & Mahmassani Shih & Mahmassani Networ parameters This This This Mandl B&M1 B&M2 B&M3 S&M1 S&M2 S&M3 Study study study 0-ansfer ips(%) or-less ansfer ip(%) or-less ansfer ip(%) Unsatisfied ip(%) No. of route Total avel Time 219, , , , , , , , , ,287 (prs-) In-vehicle Travel time 177, , , , , , , , , ,340 (prs-) Waiting time (prs-) 18,144 14,944 20,920 27,064 22,804 14,944 19,726 24,175 19,726 14,944 Transfer time(prs-) 23,500 7,003 16,650 15,600 14,800 7,003 13,550 9,550 13,550 7,003 Fleet size (veh) Table 1. Benchmar problem result comparison References [1] Baaj, "The Transit Networ design Problem:An AI-based approach", Ph.D. Dissertation, The University of Texas at Austin, [2] Shih, Mao-Chang et al.,"a Design Methodology for Bus Transit Networs with Coordinated Operaions", SWUTC /94 / , Center for Transportation, Bureau of Engineering Research, the University of Texas at Austin, [3] Spiess, H. and Florian, M.(1989), "Optimal Sategies: A New Assignment Model for Transit Networs," Transportation Research B,Vol.Ⅲ,No.2,p [4] Wei fan et al, "Optimal Transit Route Networ Design Problem",SWUTC /04 / , Center for Transportation, Bureau of Engineering Research, the University of Texas at Austin, [5] Y. Israeli and A. Ceder, "Public Transportation Assignment with passenger Sategies for Overlapping Route Choice', In Transportation and Traffic Theory(13th ISTTT), pp , 1996, [6] Y. J. Lee, Analysis and Optimization of Transit Networ Design with Integrated Routing and Scheduling, ph.d Dissertation, University. of Pennsylvania, 1998.