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Homero Larraín - The limited stop bus service design problem with stochastic passenger assignment

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Homero Larraín - The limited stop bus service design problem with stochastic passenger assignment

  1. 1. The limited-stop bus service design problem with stochastic passenger assignment Homero Larrain Pontificia Universidad Católica de Chile
  2. 2. The results presented in this webinar are taken from “Un algoritmo bi-nivel de diseño de servicios limited- stop con asignación determinística y estocástica”, a thesis by Guillermo Soto, developed in co-guidance with Juan Carlos Muñoz. Acknowledgement
  3. 3. BRT and limited-stop services The limited-stop service design problem New ideas for the LSDP Testing the new ideas Conclusions
  4. 4. Bus Rapid Transit (BRT) is a high-quality bus-based transit system that delivers fast, comfortable, and cost-effective services at metro-level capacities. It does this through the provision of dedicated lanes, with busways and iconic stations typically aligned to the center of the road, off-board fare collection, and fast and frequent operations. Institute for Transport and Development Policy What is BRT?
  5. 5. Bus Rapid Transit (BRT) is a high-quality bus-based transit system that delivers fast, comfortable, and cost-effective services at metro-level capacities. It does this through the provision of dedicated lanes, with busways and iconic stations typically aligned to the center of the road, off-board fare collection, and fast and frequent operations. Institute for Transport and Development Policy What is BRT?
  6. 6. Bus Rapid Transit (BRT) is a high-quality bus-based transit system that delivers fast, comfortable, and cost-effective services at metro-level capacities. It does this through the provision of dedicated lanes, with busways and iconic stations typically aligned to the center of the road, off-board fare collection, and fast and frequent operations. Institute for Transport and Development Policy What is BRT?
  7. 7. What is BRT? Limited-stop services are a key element in a well designed BRT system.
  8. 8. :):( :( Travel times Waiting times Transfers Operator costs :) :( :( :) Limited-stop services
  9. 9. source: brtdata.org, 2007 42 countries 165 cities 358 corridors Limited-stop services around the world
  10. 10. Express services in the literature Case-study oriented works: Ercolano (1984), Silverman (1998), Tétreault and El- Geneidy (2010), El-Geneidy and Surprenant-Legault (2010), Scortia (2010). Design models: Jordan and Turnquist (1979), Furth (1986), Leiva et al. (2010), Larrain et al. (2010, 2015), Sun et al. (2008), Chen et al. (2012), Chiraphadhanakul y Barnhart (2013).
  11. 11. BRT and limited-stop services The limited-stop service design problem New ideas for the LSDP Testing the new ideas Conclusions
  12. 12. Given a public transport corridor and a trip demand matrix for a period of time, the objective of the problem is to define a set of services (characterized by the stops they serve and omit) and their operational frequencies, so as to minimize the costs of the operator and the users. It is assumed that frequencies of the system are high. Thus, there is no scheduling involved, just frequency setting. The limited-stop service design problem
  13. 13. Designing limited-stop services is a challenging problem: 1. The problem is non-linear: operator costs are directly proportional to service frequencies, waiting costs are inversely proportional to them. 2. Binary decisions are involved: to serve or skip a stop in a particular service. 3. Passenger assignment is not trivial: it depends on the level of service of the set of available options for a trip, i.e., adding stops to a service might make it unattractive. Main challenges
  14. 14. Our previous research Leiva et al. (2010) introduced an algorithm that solves the express service design problem over a corridor: • There is an a priori set of candidate services. • Passengers are allowed to transfer. • Passenger assignment is deterministic. • Bus capacity is taken into account via a greedy heuristic. Larrain et al. (2013) extended this work: • Services are generated heuristically. • An algorithm for networks is proposed. • Many other improvements were implemented.
  15. 15. LSDP CFOAP LSGP FOAP LSDP: Limited-Stop service design problem. CFOAP: Capacitated frequency optimization and assignment problem. LSGP: Limited-Stop service generation problem. FOAP: Frequency optimization and assignment problem. Problem framework The problem at the core of the LSDP is to optimize the frequencies of a set of services, while predicting passenger behavior (i.e., which services they will use).
  16. 16. … … … … … … f1 f2 f3 f4 fn min 𝑓 𝑙,𝑓𝑙 𝑠 ,𝑉𝑠 𝑤 𝑙∈ℒ 𝑐𝑙 𝑓𝑙 + 𝜃 𝑊𝑇 𝑤∈𝒲 𝑠∈𝑆 𝑉𝑠 𝑤 𝜆 𝑙∈𝒮 𝑓𝑙 𝑠 + 𝜃 𝑇𝑇 𝑤∈𝒲 𝑠∈𝒮 𝑉𝑠 𝑤 𝑙∈ℒ 𝑡𝑙 𝑠 𝑓𝑙 𝑠 𝑙∈ℒ 𝑓𝑙 𝑠 + 𝜃 𝑇𝑟 𝑤∈𝒲 𝑇 𝑤 ′ − 𝑇 𝑤 [operator costs + waiting time + travel time + transfers] s.t.: non-negativity, flow conservation, bus conservation, frequency bounds. Objective: Minimizing social costs Frequency optimization and assignment problem (FOAP)
  17. 17. Limitations of the current approach The algorithms by Leiva and Larrain solve the FOAP using a commercial solver. Their current approach comes with the following limitations: 1. The associated MINLP problem can only be solved reasonably for instances of limited size. 2. Deterministic behavior leads to an all-or-nothing assignment, which can be unrealistic, and might make the problem even harder to solve. 3. Passenger assignment comes from minimizing social cost. This is valid only in absence of capacity constraints, so a greedy heuristic is implemented for the CFOAP.
  18. 18. BRT and limited-stop services The limited-stop service design problem New ideas for the LSDP Testing the new ideas Conclusions
  19. 19. Proposed solutions for the limitations 1. The associated MINLP problem can only be solved reasonably for limited sized instances. Implement a bi-level solution approach for the FOAP. 2. Deterministic behavior leads to an all-or-nothing assignment, which can be unrealistic, and might make the problem even harder to solve. Model passenger assignment as a stochastic process. 3. Passenger assignment comes from minimizing social cost. This is valid only in absence of capacity constraints, so a greedy heuristic is implemented for the CFOAP. Solve the capacity problem using a GRASP algorithm.
  20. 20. A bi-level solution approach for the FOAP. 1.
  21. 21. FOAP Frequency optimization and passenger assignment problem Frequency optimization Passenger assignment A bi-level approach for the FOAP Given a passenger assignment (i.e., the demand for each service), what are the optimal frequencies? For a set of given frequencies, how will passengers assign to available services?
  22. 22. LSDP CFOAP LSGP FOAP FOP PAP LSDP: Limited-Stop service design problem. CFOAP: Capacitated frequency optimization and assignment problem. LSGP: Limited-Stop service generation problem. FOAP: Frequency optimization and assignment problem. FOP: Frequency optimization problem. PAP: Assignment problem. A bi-level approach for the FOAP By separating the problem into frequency optimization and passenger assignment, we can solve bigger instances, and also work with different behavioral models.
  23. 23. Stochastic passenger assignment. 2.
  24. 24. Deterministic assignment: an overview You are at stop A waiting for a bus to go to stop B. The regular service shows up first. ¿Should you take it? A B Regular service: 𝑡 𝑟 = 60𝑚𝑖𝑛, 𝑓𝑟 = 10𝑣𝑒ℎ/ℎ Express service: 𝑡 𝑒 = 50𝑚𝑖𝑛, 𝑓𝑒 = 6𝑣𝑒ℎ/ℎ Option 1: Take it. Waiting time: 0 Travel time: 60𝑚𝑖𝑛 Option 2: Leave it. Waiting time: 5𝑚𝑖𝑛 Travel time: 50𝑚𝑖𝑛 For every O/D pair in our corridor we can solve a similar problem, where we find the attractive set of services to perform that trip.
  25. 25. Deterministic assignment: an overview The assignment process consists of three steps: 1. Finding the attractive services and the expected travel time for every O/D pair. 2. On the resulting network, assign trips to their shortest paths. 3. Compute the ridership for each service.
  26. 26. Deterministic assignment: an overview 1. Finding the attractive services and the expected travel time for every O/D pair. 𝑙1: 𝑡𝑡1, 𝑓1 𝑙2: 𝑡𝑡2, 𝑓2 𝑙 𝑛: 𝑡𝑡 𝑛, 𝑓𝑛 𝑡𝑡1 ≤ 𝑡𝑡2 ≤ ⋯ ≤ 𝑡𝑡 𝑛 … The attractive set of lines for A-E, 𝑆𝐴𝐸, is the set that minimizes the expected travel time: 𝐸𝑇𝑇𝑆 𝑤 𝑤 = 𝜙 ∙ 𝑘 𝑙∈𝑆 𝑤 𝑓𝑙 + 𝑙∈𝑆 𝑤 𝑡𝑡𝑙 𝑤 ∙ 𝑓𝑙 𝑙∈𝑆 𝑤 𝑓𝑙 [expected waiting + travel times]
  27. 27. Deterministic assignment: an overview 1. Finding the attractive services and the expected travel time for every O/D pair. 𝑙1: 𝑡𝑡1, 𝑓1 𝑙2: 𝑡𝑡2, 𝑓2 𝑙 𝑛: 𝑡𝑡 𝑛, 𝑓𝑛 𝑡𝑡1 ≤ 𝑡𝑡2 ≤ ⋯ ≤ 𝑡𝑡 𝑛 … 𝐸𝑇𝑇𝐴𝐸, 𝑆𝐴𝐸
  28. 28. Deterministic assignment: an overview 2. On the resulting network, assign trips to their shortest paths. 𝑙1: 𝑡𝑡1, 𝑓1 𝑙2: 𝑡𝑡2, 𝑓2 𝑙 𝑛: 𝑡𝑡 𝑛, 𝑓𝑛 𝑡𝑡1 ≤ 𝑡𝑡2 ≤ ⋯ ≤ 𝑡𝑡 𝑛 … 𝐸𝑇𝑇𝐴𝐸, 𝑆𝐴𝐸
  29. 29. Deterministic assignment: an overview 3. Compute the ridership for each service. 100 +50 +50 +50+50 80 +30+30 +20 etc…
  30. 30. There are many ways to model stochasticity in passenger assignment on this problem. In our model we introduce it in two ways: • Stochastic choice for the set of attractive lines (step 1). • Stochastic route choice (step 2). Stochastic assignment
  31. 31. Selection of the set 𝑆 𝑤 is no longer deterministic. We model this process using a Logit model. The cost of the O/D pair is now represented by its expected maximum utility. 𝑙1: 𝑡𝑡1, 𝑓1 𝑙2: 𝑡𝑡2, 𝑓2 𝑙 𝑛: 𝑡𝑡 𝑛, 𝑓𝑛 𝐸𝑀𝑈 = − 1 𝜃 ln 𝑙=1 𝑛 exp −𝜃 ∙ 𝐸𝑇𝑇 1,…,𝑖 … 𝐸𝑀𝑈 𝑤 Pr 1 Pr 1,2 Pr 1,2, … , 𝑛 … 𝐸𝑇𝑇{1} 𝐸𝑇𝑇{1,2} 𝐸𝑇𝑇{1,2,…,𝑛} Pr 1, … , 𝑖 = exp −𝜃 ∙ 𝐸𝑇𝑇 1,…,𝑖 𝑗=1 𝑛 exp −𝜃 ∙ 𝐸𝑇𝑇 1,…,𝑗 Stochastic assignment: attractive services
  32. 32. To model route choice, now we use Dial’s algorithm (1971). 𝐸𝑀𝑈 1,3 𝐸𝑀𝑈 2,4 𝐸𝑀𝑈 1,2 𝐸𝑀𝑈 2,3 𝐸𝑀𝑈 3,4 𝐸𝑀𝑈 1,4 𝑇 1,4 𝑇 1,4 Stochastic assignment: route choice 𝑇 1,4 ∙ Pr 𝑝1 𝑇 1,4 ∙ Pr 𝑝2 𝑇 1,4 ∙ Pr 𝑝3 Total demand for each service can be obtained like before, but taking into account the probability of choosing each possible subset 𝑆 𝑤.
  33. 33. A GRASP capacity algorithm. 3.
  34. 34. Dealing with bus capacity CFOAP greedy heuristic 1 solve an instance of the FOAP, obtaining 𝑓 𝑛 . 2 while there is a line 𝑙 that shows a capacity deficit: 3 Select 𝑙′ as the line with the greater capacity deficit. 4 Increase the lower bound for 𝑓𝑙′. 5 Re-solve the FOAP, and update 𝑓 𝑛+1 . This is a greedy heuristic that can be easily extended to a GRASP algorithm (but we need a fast solution for the FOAP). Leiva / Larrain use the following heuristic for solving the capacitated design problem:
  35. 35. Dealing with bus capacity CFOAP greedy heuristic 1 solve an instance of the FOAP, obtaining 𝑓 𝑛 . 2 while there is a line 𝑙 that shows a capacity deficit: 3 Select 𝑙′ as the line with the greater capacity deficit. 4 Increase the lower bound for 𝑓𝑙′. 5 Re-solve the FOAP, and update 𝑓 𝑛+1 . CFOAP GRASP heuristic 1 solve an instance of the FOAP, obtaining 𝑓 𝑛 . 2 while there is a line 𝑙 that shows a capacity deficit: 3 Select 𝑙′ randomly from the lines in deficit. 4 Increase the lower bound for 𝑓𝑙′. 5 Re-solve the FOAP, and update 𝑓 𝑛+1 .
  36. 36. BRT and limited-stop services The limited-stop service design problem New ideas for the LSDP Testing the new ideas Conclusions
  37. 37. Case study We optimized the design for three corridors: Pajaritos and Grecia in Santiago, and Caracas in Bogotá. For each case we generated a 20, 40 and 80 stop version of the O-D matrices, thus defining 9 scenarios. 0 5000 10000 15000 20000 T 3 6 9 12 15 18 21 24 27 30 33 36 39 41 44 47 50 53 56 59 62 65 68 71 74 77 80 Pajaritos Grecia Caracas 1 40 41 80 Demand(pax/h) Stops 20.000 15.000 10.000 5.000 0 Corridor loads
  38. 38. The experiment Algorithm Optimization strategy Passenger assignment Capacity heuristic Transfers allowed A0 Simultaneous Deterministic Greedy No D / g / N Bi-level Deterministic Greedy No D / G / N Bi-level Deterministic GRASP No D / g / T Bi-level Deterministic Greedy Yes D / G / T Bi-level Deterministic GRASP Yes S / g / T Bi-level Stochastic Greedy Yes S / G / T Bi-level Stochastic GRASP Yes These algorithms were applied on each of the nine scenarios described before.
  39. 39. • We confirm that by separating the problem we don’t lose the quality of the solution. • Transfers seem to have a bigger impact as demand grows. • Greedy algorithms sometimes converge to suboptimal solutions, but in general they perform reasonably well. • In the stochastic case there are some unexpected trends, likely due to suboptimality of the solutions. However, the algorithm still manages to beat the all-stop solution. General results: savings CTC ($/h) Corrected percentage savings (CPS) Scenario All-stop A0 D / g / N D / G / N D / g / T D / G / T S / g / T S / G / T P20 16,920 48.5% 48.5% 48.5% 48.5% 48.5% 65.6% 65.6% P40 33,847 56.2% 56.5% 56.5% 56.5% 56.5% 48.8% 48.8% P80 68,353 63.7% 63.8% 63.8% 63.8% 63.8% 48.2% ** G20 18,070 22.2% 22.2% 22.2% 22.2% 22.2% 48.7% 50.0% G40 36,589 31.3% 31.2% 31.2% 31.3% 31.3% 27.6% 27.6% G80 73,873 35.7% 35.6% 35.6% 38.0% 38.0% 22.8% ** C20 22,142 13.2% 17.8% 17.8% 14.7% 18.3% 0.0%* 0.0%* C40 36,540 30.2% 31.3% 31.6% 34.4% 34.5% 1.6% 2.6% C80 172,995 79.4% 79.3% 79.6% 81.4% 81.4% 71.0% **
  40. 40. The effect of express services is much greater when transfer costs are reduced! The impact of transfer costs Corrected percentage savings (CPS) Scenario (Transfer) D / G / N No transfers D / G / T* Transfers D / G / B* Bidirectional D / G / FT* Free Transf. D / G / FB* Free Bid. Tr. P20 48,50% 48.5% (0) 48.5% (0) 48.7% (430) 48.7% (430) P40 56,50% 56.5% (12) 56.5% (18) 57.3% (1169) 57.5% (1,882) P80 63,80% 63.8% (511) 64.2% (2,192) 65.5% (10655) 66.9% (17,459) G20 22,20% 22.2% (0) 22.2% (0) 23.6% (2100) 23.6% (2,100) G40 31,20% 31.3% (187) 31.3% (412) 35.8% (7445) 35.8% (8,221) G80 35,60% 38.0% (3041) 39.6% (7,074) 45.1% (16331) 47.6% (29,391) C20 17,80% 18.3% (2085) 18.3% (2,085) 22.9% (11718) 22.9% (11,718) C40 31,60% 34.5% (6786) 34.5% (6,786) 39.5% (19731) 39.5% (19,274) C80 79,60% 81.4% (15950) 81.5% (17,383) 83.0% (31278) 83% (32,208) * Number of transfers is given in parenthesis.
  41. 41. Solution times The bi-level approach finds good solutions in just a fraction of the time! Scenario A0 D / g / N D / G / N* D / g / T D / G / T* S / g / T S / G / T* P20 1 3 59 (10) 4 88 (10) 9 189 (10) P40 32 11 232 (10) 11 378 (10) 161 973 (5) P80 984 48 1,318 (10) 8 197 (10) 1,912 ** G20 3 28 222 (10) 9 267 (10) 127 1005 (10) G40 31 4 86 (10) 3 79 (10) 40 349 (5) G80 169 28 411 (10) 9 235 (10) 9,289 ** C20 17 10 128 (10) 3 135 (10) 10 135 (10) C40 219 59 2,127 (10) 22 191 (10) 298 1573 (5) C80 1,382 149 4,427 (10) 110 1364 (10) 17,241 ** * Number of iterations is given in parenthesis.
  42. 42. The impact of stochastic assignment 0 20 40 60 80 100 120 140 T 3 6 9 12 15 18 21 24 27 30 33 36 39 41 44 47 50 53 56 59 62 65 68 71 74 77 80 0 20 40 60 80 100 120 140 T 3 6 9 12 15 18 21 24 27 30 33 36 39 41 44 47 50 53 56 59 62 65 68 71 74 77 80 Bus capacity All Stop All stop Deterministic assignment Stochastic assignment Load(pax/bus) 1 40 41 80 1 40 41 80 Bus load per service Stochastic assignment gives more robust solutions, spreading demand among services instead of saturating a few of them.
  43. 43. What if we got assignment wrong?... What if we design for deterministic passenger assignment, but passengers really behave stochastically?
  44. 44. 0 120 240 360 480 600 T 3 6 9 12 15 18 21 24 27 30 33 36 39 41 44 47 50 53 56 59 62 65 68 71 74 77 80 Deterministic design facing stochastic behavior All Stop All stop L17 L36 L23 1 40 41 80 Bus capacity Load(pax/bus) What if we got assignment wrong?... Deterministic assignment underestimates regular-like service demands!
  45. 45. BRT and limited-stop services The limited-stop service design problem New ideas for the LSDP Testing the new ideas Conclusions
  46. 46. • We formally introduce the Limited-stop Service Design Problem and propose a solution framework for it. • We greatly improve solution times for the deterministic version of the Capacitated Frequency Optimization and Assignment Problem, which solves the LSDP for a given set of services. • Stochastic assignment leads to more realistic and robust solutions, but makes the problem harder to solve. The bi- level approach finds solutions for this variant of the problem, but there is room for improvement. Conclusions
  47. 47. • Allowing for transfers during service design can lead to better solutions. We also can conclude that reducing the nuisance associated with transfers can improve the performance of limited-stop services. • Future research should tackle the issue of capacity at a bus stop level, which is an active constraint in existing systems, but adds a new layer of complexity to the problem. • Another future avenue for research consists in solving the design problem at a BRT network level, which would allow to better consider during the design phase the complex travel patterns of the system. Conclusions
  48. 48. The limited-stop bus service design problem with stochastic passenger assignment Homero Larrain Pontificia Universidad Católica de Chile

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