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A car sharing auction with temporal-spatial OD connection conditions

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A car sharing auction with temporal-spatial OD connection conditions

  1. 1. A car sharing auction with temporal-spatial OD connection conditions Yusuke Hara (The University of Tokyo) Eiji Hato (The University of Tokyo) 22nd International Symposium on Transportation and Traffic Theory @ Northwestern University
  2. 2. A core problem of Transportation and Traffic Theory 1. Bottleneck model (Vickrey, 1969) – travel behavior: departure time choice – policy: congestion charging 2. Traffic Assignment – UE (Beckmann,1956), SUE (Daganzo and Sheffi, 1977), Hyperpath (Spiess and Florian, 1989; Bell, 2009) – travel behavior: route choice – policy: pricing of links for system optimum (Henderson, 1985) 3. Tradable permit scheme – Akamatsu (2007), Yang and Wang (2011), Wu et al. (2012), Nie and Yin (2013), Wada and Akamatsu (2013) – travel behavior: route choice and departure time choice – policy: market mechanism for optimal allocation 2 The Negative Externality of people’s trips is an important problem for transportation system.
  3. 3. What is the cause of the negative externality? 3 1. Capacity of transportation system 2. Users use the same transport facility simultaneously Road network Mobility service t = 1 t = 2 t = 3 t = 1 t = 2 t = 3 Link capacity of each road link is fixed. Service capacity of each OD and time slot depends on vehicle paths. μ Link capacity (aggregate capacity) Vehicle capacity (disaggregate capacity)
  4. 4. What is the cause of the negative externality? 4 1. Capacity of transportation system 2. Users use the same transport facility simultaneously 3. Flexible OD of mobility services generates negative/positive externalities. t = 1 t = 2 t = 3 t = 1 t = 2 negative externality positive externality “Sharing a vehicle by more than one person” is not essential. “Connecting trips temporally and spatially” is essential. The important point of mobility sharing
  5. 5. Our Concept 5 bottleneck network service with “knots” link capacity single OD aggregate capacity multi aggregate OD disaggregate capacity multi disaggregate OD temporal-spatial connection condition Designing mobility sharing auction mechanism considering temporal-spatial OD connection condition. Research Objective = “knot”
  6. 6. Our approach 6 Fundamental Theory for mobility services with “knots” Service Implementation in real world and Travel Behavior Analysis for tradable permit Transportation ECO Point System Probe Person System Bicycle Tradable Permits System Bicycle Sharing System GPS data travel diary data give point use by point settlement booking use by permit trade by point settlement stock management Today’s presentation Hara and Hato (Transportation, 2017)
  7. 7. Setting1: the Assumption of Transportation Services 7 We don’t consider the delay effect. 1 2 3 4 時間 空間ネットワーク 利用時間枠 t = 1 利用時間枠 t = 2 利用時間枠 t = 3 利用時間枠 t = 4 t = 1 t = 2 t = 3 t = T : port node : OD pair pq IiÎ Tt Î Lpq Î : user i : time slot t Nqp Î, We assume the following statements. 1. Mobility sharing means car sharing in this study. 2. A trip can be done in a time slot. 3. There is no delay. 4. The number of vehicle is μ and each vehicle can move independently. temporal-spatial network time slot time slot time slot time slot time slot time
  8. 8. Setting 2: Supplier, Users and the Capacity limit 8 We assume that service supplier is the player to maximize social welfare by operating vehicle effectively. The supplier tackles an unbalanced demand problem between ports by allocating users to permits satisfying temporal-spatial OD connected condition at all time slot. the setting of service supplier vehicle capacity limit μ Vehicle capacity limit means the number of vehicles service supplier has. Service supplier can issue permits depending on the vehicle capacity limit. { }1,0)( Îtxi pq is a discrete variable. If user i is allocated to a permit to use vehicle between OD pq at time slot t, this variable is 1. Otherwise this variable is 0. Users have the value of permits and the value of each permit is different by time slot and OD pair. Users determine the value by their OD demand, their desired usage timeslot, their value of time and so on. the setting of users and users’ value of permit
  9. 9. Setting 3: Temporal-Spatial OD connection Condition 9 時間 空間ネットワーク 利用時間枠 t )1( -txi pq 利用時間枠 t ‒1 )(txi qp åå - i p i pq tx )1( ååi p i qp tx )( q q Meeting temporal-spatial OD connection condition is satisfying the following equation for any node q. time time slot time slot
  10. 10. Example of the Setting User ID OD value (t = 1) value (t = 2) value (t = 3) A 1→2 90 80 70 B 1→2 60 70 80 C 2→1 70 80 70 D 2→3 70 60 50 E 3→1 50 60 70 10 Users’ demand and value of permit Example network port 1 port 2 port 3 A trip-connected path allocation {A, D, E} A trip-connected path allocation {A, C, B} Sum of values (Social Welfare) is 220. t=1 t=2 t=3 t=1 t=2 t=3 A D E A C B Sum of values (Social Welfare) is 250.
  11. 11. Permits Allocation of Social Optimum • optimization problem to maximize the sum of users’ values[SO] 11 single demand condition capacity limit temporal-spatial OD connection each user’s permits allocation If users bid their true values, service supplier only has to solve this optimization problem [SO] and social optimum is achieved. However, if users bid their false values strategically, it is difficult to maximize social welfare. Therefore, we need a mechanism to make users bid true values.
  12. 12. VCG Mechanism for Mobility Sharing 12 Vickrey–Clarke–Groves Mechanism satisfy efficiency and strategy-proofness. We extend VCG mechanism for mobility sharing permits. 1. All users bid on the permits, which becomes the users’ demand. 2. Auctioneers decide the allocation of permits in order to maximize the sum of bidding values under temporal-spatial OD connection conditions and capacity limit conditions 3. Winners must pay for the permits and the price is the winner’s externality, which is the decrease in optimal social welfare when she is included in the auction. The price is called “Vickrey payments”. VCG mechanism for mobility sharing permit Under this mechanism, users have an incentive to bid their true values because they cannot get the larger utilities by telling a lie than the utilities by telling truth. Therefore, supplier can solve the optimization problem and the social optimum is achieved.
  13. 13. The example of Vickrey payments 13 A C B Social Welfare = 250 Social Welfare except A = 160 User ID OD value (t = 1) value (t = 2) value (t = 3) A 1→2 90 80 70 B 1→2 60 70 80 C 2→1 70 80 70 D 2→3 70 60 50 E 3→1 50 60 70 90 80 80 D E B Social Welfare without A = 210 70 60 80 Maximum Social Welfare Allocation Maximum Social Welfare Allocation without A 𝑃" 𝒗 = 𝑊 0, 𝒗(" − 𝑊(" (𝒗) The payment of user i Maximum Social Welfare without i in the bidders set Maximum Social Welfare except i’s value 𝑃, 𝒗 = 210 – 160 = 50 The Vickrey payment is the negative externality of user A’s usage.
  14. 14. The Solution Method of Single-Minded Bid Case • The winner determination problem [SO] is included in combinatorial optimization problems. And these problems are NP-hard in general. • First, we assume that users bid for only 1 timeslot permit. • Users will bid for the most variable timeslot permit for them. • This assumption is single-minded bids in combinatorial auction. • Computational effort is small. • In this case, we can interpret the price by the dual problem. 14
  15. 15. Linear relaxation is identical to LP 15 • In single-minded bidder setting, the social optimization problem is redefined by the following equations: subject to Since the constraint coefficient matrix is totally unimodular, the solution of the linear relaxation problem [SO-SMB] is identical to that of the LP problem [SO-SMB-LP] (see Hoffman and Kruskan (1956)) linear relaxation Solving LP problem is much easier than IP problem !
  16. 16. Vickrey payments decomposition • From the dual problem of [SO-SMB], 16 usage fee to leave p at t income to arrive at q at t+1 1) Vickrey payments are decomposed into the payment for the previous user and the income from the following user 2) Vickrey payments between the same OD pairs and time slots are the same price 3) Users leave their origin as consumers and arrive at their destination as suppliers. 4) What they consume (or supply) is the opportunity to use mobility sharing. 𝑃-.(𝑡) = 𝑎- 𝑡 − 𝑎. 𝑡 + 1 ∀𝑡 t t+1 usage fee income
  17. 17. Negative price • In this mechanism, there is negative price. • This mechanism generates the negative price and efficient allocation naturally. 札者 間帯 負条 901 U: 算機 にそ す. ポート1 ポート2 図–8 OD 需要が偏った場合 表–4 OD 需要が偏った場合の入札者集合 入札者 OD v(t = 1) v(t = 2) v(t = 3) 1 1 → 2 95 90 85 2 1 → 2 82 92 72 3 1 → 2 74 86 61 4 1 → 2 60 80 92 5 2 → 1 30 20 10 個人の支払額は P1 = (82 + 20 + 92) − (20 + 92) = 82 P5 = (95) − (95 + 92) = −92 P4 = (95 + 20 + 72) − (95 + 20) = 72 となる.これは第 1 項がその個人が入札しなかった場合 の評価値を最大とする割当て時の評価値の総和を表し ており,入札者 ID = 5 がいない場合は ID = 1 がポー. ポート1 ポート2 図–8 OD 需要が偏った場合 表–4 OD 需要が偏った場合の入札者集合 入札者 OD v(t = 1) v(t = 2) v(t = 3) 1 1 → 2 95 90 85 2 1 → 2 82 92 72 3 1 → 2 74 86 61 4 1 → 2 60 80 92 5 2 → 1 30 20 10 個人の支払額は P1 = (82 + 20 + 92) − (20 + 92) = 82 P5 = (95) − (95 + 92) = −92 P4 = (95 + 20 + 72) − (95 + 20) = 72 となる.これは第 1 項がその個人が入札しなかった場合 の評価値を最大とする割当て時の評価値の総和を表し ており,入札者 ID = 5 がいない場合は ID = 1 がポー 2000 5000 10000 20 50 100 20 10 20 10 20 10 図–7 大規模計算時間の比較 ,ポート数 50,時間帯枠数 t = 10,(3) 入札者 人,車両台数 µ = 20,ポート数 100,時間帯 の計算時間を示す.また,それぞれ非負条 約条件式の数は (1)2181,(2)5451,(3)10901 算には OS: Windowa 7 Professional, CPU: i7-2620M (2.7GHz), メモリ 8GB の計算機 に発生させた評価値・OD 分布をもとにそ ースで 10 回実施した平均値を図–7 に示す. 均 (1)0.18 [sec], (2)0.63 [sec], (3)1.81 [sec] 題ない実行時間である.勝者決定問題を解 用権価格としての Vickrey payments を計算 あるが,これも各勝者を入札者集合から抜 図–8 OD 需要が偏った場合 表–4 OD 需要が偏った場合の入札者集合 入札者 OD v(t = 1) v(t = 2) v(t = 3) 1 1 → 2 95 90 85 2 1 → 2 82 92 72 3 1 → 2 74 86 61 4 1 → 2 60 80 92 5 2 → 1 30 20 10 個人の支払額は P1 = (82 + 20 + 92) − (20 + 92) = 82 P5 = (95) − (95 + 92) = −92 P4 = (95 + 20 + 72) − (95 + 20) = 72 となる.これは第 1 項がその個人が入札しなかった場合 の評価値を最大とする割当て時の評価値の総和を表し ており,入札者 ID = 5 がいない場合は ID = 1 がポー ト 1 からポート 2 へ移動した後,車両がポート 1 へ戻 ることができないため,t = 2, 3 の時間帯に利用され ないことを表している.しかし,入札者 ID = 5 が存在 することで車両がポート 1 へ戻り,入札者 ID = 4 が利 The payment of each userusers Port 1 Port 2 17
  18. 18. The extension for activity chain • If some users want round trips, our model framework can be extended easily. 18 14 Hara and Hato / Transportation Research Procedia 00 (2016) 000–000 user assumes that only single trip is worthless and that it is variable to be assigned both first trip and Hence, only the bundle of first trip and second trip is variable. In this setting, each user i need to report vi and the duration ki ∈ N from first trip and second trip. Under the auction setting, socially optimal allocation maximizes the sum of the assigned user’s valu derived by the following optimization equation [SO-round]: max x i∈I t∈T pq∈L vi pq(t) · xi pq(t), subject to i∈I pq∈L xi pq(1) + q∈N xS qq(1) = µ, − i∈I p∈N xi pq(t − 1) − xS qq(t − 1) + i∈I p∈N xi qp(t) + xS qq(t) = 0 t = 2, . . . , T, ∀q ∈ N, t∈T pq∈L xi pq(t) ≤ 2, ∀i ∈ I xi pq(t) − xi qp(t + ki) = 0 ∀i ∈ I, ∀p, q ∈ N, ∀t ∈ {1, 2, . . . T − ki} xi pq(t) ∈ {0, 1}, ∀i ∈ I, ∀pq ∈ L, ∀t ∈ T xS qq(t) ∈ N. ∀q ∈ N, ∀t ∈ T Eqs.(37, 38, 39, 42, and 43) are equivalent to a scenario with only a one-way trip situation. The differ i i
  19. 19. The extension for activity chain • If some users want round trips, our model framework can be extended easily. 19 t∈T pq∈L xi pq(t) − xi qp(t + ki) = 0 ∀i ∈ I, ∀p, q ∈ N, ∀t ∈ {1, 2, . . . T − ki} (41) xi pq(t) ∈ {0, 1}, ∀i ∈ I, ∀pq ∈ L, ∀t ∈ T (42) xS qq(t) ∈ N. ∀q ∈ N, ∀t ∈ T (43) Eqs.(37, 38, 39, 42, and 43) are equivalent to a scenario with only a one-way trip situation. The difference is with respect to the constraint conditions Eqs.(40 and 41). These equations either satisfy both xi pq(t) = 1 and xi qp(t + ki) = 1, or both xi pq(t) = 0 and xi qp(t + ki) = 0. The optimization problem [SO-round] is more complex than [SO], but it is only added to the linear constraint conditions. Therefore, the solution method is same as the original problem. 1 2 3 time slot t = 3 time slot t = 2 time slot t = 1 Time 16 time slot t = 4 time slot t = 5 1 2 3 Time 21 20 20 16 18 17 13 Fig. 5. Winners and vehicles’ paths in the round trip case. To demonstrate the round trip case, we show the users’ bidding value example in Table 3. Thirteen users desire
  20. 20. The solution method in multiple bidding case • In the case of multiple bidding case, the LP relaxation is not guaranteed. • We propose the new solution method by combining the primal-dual algorithm and a branch and bound algorithm. • Finally, we compare the exact solution (multiple bidding), approximate solution (single-minded bidding), and First come, first serve rule. 20
  21. 21. 10 20 30 40 50 The number of time slots Fig. 6. Computational Times of numerical examples. 400 500 600 700 800400 500 600 700 800 0.0000.0010.0020.0030.0040.005 Multiple bidding auction Single-minded bidding auction First come, first served (max value choice) First come, first served (logit choice) First come, first served (random choice) Probabilitydensity Social Welfare Fig. 7. Efficiency of mobility sharing auction and first come, first serve rule. Efficiency of mobility sharing auction and first come, first serve rule 21 Exact solution (837) Approximate solution (LP relaxation) (833)
  22. 22. Conclusions and Future work • The essence of mobility sharing is the service with “knots” of trips/paths. • For satisfying temporal-spatial OD connection condition, we proposed the mobility sharing auction. • From the LP relaxation, Vickrey payment is decomposed into usage fee and income. • It means mobility sharing is the transaction on “knots” to exchange the opportunity of vehicles usage. • For future work, we need to study – Dynamic mechanism (online auction setting) – Preference elicitation mechanism for more easier bidding system because bidding all items is high cognitive cost for users – Comparison with other reservation systems and other mobility services (for example, ride sharing) 22 Thank you for your attention!

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