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Algorithms for train scheduling on a single line Laurent DAUDET 1 PhD advisor: Frdric MEUNIER 1 1 CERMICS, Centre dEnseignement et de Recherche en Mathmatiques et Calcul Scientifique, ENPC December 22nd, 2017 Laurent DAUDET PhD Defense

  1. One-hour schedules maximizing HGV shuttles Model The model Variables � � : scheduled departure times of all trains. d = d Eur j , d Fr j , d PAX , d HGV j j j : j th departure time of train of type A. → d A n : number of scheduled HGV shuttles. Mathematical model Max ( d , n ) ∈ X n where X = set of constraints → can be expressed with linear constraints (only non immediate constraint: security headway [Serafini and Ukovich, 1989]). ⇒ Mixed Integer Linear Program Solved by commercial solver CPLEX. Laurent DAUDET PhD Defense December 22nd, 2017 8 / 40

  2. One-hour schedules maximizing HGV shuttles Numerical results A current schedule Instance: 4 Eurostars, 5 PAX shuttles, and 1 freight train. Laurent DAUDET PhD Defense December 22nd, 2017 9 / 40

  3. One-hour schedules maximizing HGV shuttles Numerical results A current schedule Instance: 4 Eurostars, 5 PAX shuttles, and 1 freight train. ⇒ Maximum 4 HGV shuttles in the schedule Laurent DAUDET PhD Defense December 22nd, 2017 9 / 40

  4. One-hour schedules maximizing HGV shuttles Numerical results A current schedule Instance: 4 Eurostars, 5 PAX shuttles, and 1 freight train. ⇒ Maximum 4 HGV shuttles in the schedule Laurent DAUDET PhD Defense December 22nd, 2017 9 / 40

  5. One-hour schedules maximizing HGV shuttles Numerical results Some schedule improvements Parameters T : length of the period. L : 12-minute time-window. η : full minute discretization. C Eur : 30-minute gap between grouped Eurostars. Laurent DAUDET PhD Defense December 22nd, 2017 10 / 40

  6. One-hour schedules maximizing HGV shuttles Numerical results Some schedule improvements Parameters T : length of the period. L : 12-minute time-window. η : full minute discretization. C Eur : 30-minute gap between grouped Eurostars. Constraint relaxed Instance Improvement Length cyclic period T : 1 h → 4 h Up to 25% Loading platforms L : 12 min → 0 min Up to 50% Full minute discretization η : 1 min → 1 s Up to 33% Agreements with Eurostar C Eur : 30 min → [27 min-33 min] Up to 14% Laurent DAUDET PhD Defense December 22nd, 2017 10 / 40

  7. One-hour schedules maximizing HGV shuttles Numerical results Some other problems Compute one-hour schedules with minimum delays. Compute one-hour schedules with maximum number of HGV shuttles and minimum delays. → Stochastic Optimization, Sample Average Approximation. Laurent DAUDET PhD Defense December 22nd, 2017 11 / 40

  8. One-hour schedules maximizing HGV shuttles Numerical results Some other problems Compute one-hour schedules with minimum delays. Compute one-hour schedules with maximum number of HGV shuttles and minimum delays. → Stochastic Optimization, Sample Average Approximation. Laurent DAUDET PhD Defense December 22nd, 2017 11 / 40

  9. Joint scheduling and pricing problem Problem General context 1 One-hour schedules maximizing HGV shuttles 2 Joint scheduling and pricing problem 3 Minimizing the waiting time for a one-way shuttle service 4 Laurent DAUDET PhD Defense December 22nd, 2017 11 / 40

  10. Joint scheduling and pricing problem Problem Why such a problem? Departures and prices computed jointly in airline companies. → Increase of customers’ satisfaction and company’s revenue. Same objective for rail transportation. Toy problem to challenge this idea. Laurent DAUDET PhD Defense December 22nd, 2017 12 / 40

  11. Joint scheduling and pricing problem Problem Why such a problem? Departures and prices computed jointly in airline companies. → Increase of customers’ satisfaction and company’s revenue. Same objective for rail transportation. Toy problem to challenge this idea. Laurent DAUDET PhD Defense December 22nd, 2017 12 / 40

  12. Joint scheduling and pricing problem Problem Why such a problem? Departures and prices computed jointly in airline companies. → Increase of customers’ satisfaction and company’s revenue. Same objective for rail transportation. Toy problem to challenge this idea. Laurent DAUDET PhD Defense December 22nd, 2017 12 / 40

  13. Joint scheduling and pricing problem Problem Why such a problem? Departures and prices computed jointly in airline companies. → Increase of customers’ satisfaction and company’s revenue. Same objective for rail transportation. Toy problem to challenge this idea. Laurent DAUDET PhD Defense December 22nd, 2017 12 / 40

  14. Joint scheduling and pricing problem Problem The problem (1/2) One-way trip. Laurent DAUDET PhD Defense December 22nd, 2017 13 / 40

  15. Joint scheduling and pricing problem Problem The problem (1/2) One-way trip. Company wants to fix departures d of S trains and prices p . Each train has finite capacity C . Q customers want to purchase tickets for this trip. Buy tickets that satisfies them the most, or leave without purchasing. Laurent DAUDET PhD Defense December 22nd, 2017 13 / 40

  16. Joint scheduling and pricing problem Problem The problem (1/2) One-way trip. Company wants to fix departures d of S trains and prices p . Each train has finite capacity C . Q customers want to purchase tickets for this trip. Buy tickets that satisfies them the most, or leave without purchasing. Laurent DAUDET PhD Defense December 22nd, 2017 13 / 40

  17. Joint scheduling and pricing problem Problem The problem (1/2) One-way trip. Company wants to fix departures d of S trains and prices p . Each train has finite capacity C . Q customers want to purchase tickets for this trip. Buy tickets that satisfy them the most, or leave without purchasing. Laurent DAUDET PhD Defense December 22nd, 2017 13 / 40

  18. Joint scheduling and pricing problem Problem The problem (1/2) One-way trip. Company wants to fix departures d of S trains and prices p . Each train has finite capacity C . Q customers want to purchase tickets for this trip. → Buy tickets that satisfy them the most, or leave without purchasing. Laurent DAUDET PhD Defense December 22nd, 2017 13 / 40

  19. Joint scheduling and pricing problem Problem The problem (1/2) One-way trip. Company wants to fix departures d of S trains and prices p . Each train has finite capacity C . Q customers want to purchase tickets for this trip. → Buy tickets that satisfy them the most, or leave without purchasing. Objective Maximize the revenue of the company. Laurent DAUDET PhD Defense December 22nd, 2017 13 / 40

  20. Joint scheduling and pricing problem Problem The problem (2/2) Each customer i has a preferred departure time: random variable χ i belongs to economic class b i (e.g. business, tourist, low-cost, ...) “value of time” v b for economic class b Laurent DAUDET PhD Defense December 22nd, 2017 14 / 40

  21. Joint scheduling and pricing problem Problem The problem (2/2) Each customer i has a preferred departure time: random variable χ i belongs to economic class b i (e.g. business, tourist, low-cost, ...) “value of time” v b for economic class b Laurent DAUDET PhD Defense December 22nd, 2017 14 / 40

  22. Joint scheduling and pricing problem Problem The problem (2/2) Each customer i has a preferred departure time: random variable χ i belongs to economic class b i (e.g. business, tourist, low-cost, ...) “value of time” v b for economic class b Laurent DAUDET PhD Defense December 22nd, 2017 14 / 40

  23. Joint scheduling and pricing problem Problem The problem (2/2) Each customer i has a preferred departure time: random variable χ i belongs to economic class b i (e.g. business, tourist, ...) “value of time” v b for economic class b We assume v 1 ≤ v 2 ≤ · · · Customers of class 1 make their choice first, then 2, ... Laurent DAUDET PhD Defense December 22nd, 2017 14 / 40

  24. Joint scheduling and pricing problem Problem The problem (2/2) Each customer i has a preferred departure time: random variable χ i belongs to economic class b i (e.g. business, tourist, ...) “value of time” v b for economic class b We assume v 1 ≤ v 2 ≤ · · · Customers of class 1 make their choice first, then 2, ... Discrete choice model Each customer i and product j (departure d j at price p j ) → Random utility U ij ( d , p ) representing satisfaction. Laurent DAUDET PhD Defense December 22nd, 2017 14 / 40

  25. Joint scheduling and pricing problem Model The model (1/2) Denote by ξ vector representing uncertainty. Revenue of company → R ( d , p , ξ ) where ξ has been revealed. → Easy to compute (simulation, Linear Programming). Objective function → f ( d , p ) = E [ R ( d , p , ξ )] . → Remark: f ( d , p ) well defined (for all ( d , p ) ∈ X , R ( d , p , · ) measurable and E [ R ( d , p , ξ )] < ∞ ) and Var [ R ( d , p , ξ )] < ∞ . Laurent DAUDET PhD Defense December 22nd, 2017 15 / 40

  26. Joint scheduling and pricing problem Model The model (1/2) Denote by ξ vector representing uncertainty. Revenue of company → R ( d , p , ξ ) where ξ has been revealed. → Easy to compute (simulation, Linear Programming). Objective function → f ( d , p ) = E [ R ( d , p , ξ )] . → Remark: f ( d , p ) well defined (for all ( d , p ) ∈ X , R ( d , p , · ) measurable and E [ R ( d , p , ξ )] < ∞ ) and Var [ R ( d , p , ξ )] < ∞ . Laurent DAUDET PhD Defense December 22nd, 2017 15 / 40

  27. Joint scheduling and pricing problem Model The model (1/2) Denote by ξ vector representing uncertainty. Revenue of company → R ( d , p , ξ ) where ξ has been revealed. → Easy to compute (simulation, Linear Programming). Objective function → f ( d , p ) = E [ R ( d , p , ξ )] . → Remark: f ( d , p ) well defined (for all ( d , p ) ∈ X , R ( d , p , · ) measurable and E [ R ( d , p , ξ )] < ∞ ) and Var [ R ( d , p , ξ )] < ∞ . Laurent DAUDET PhD Defense December 22nd, 2017 15 / 40

  28. Joint scheduling and pricing problem Model The model (1/2) Denote by ξ vector representing uncertainty. Revenue of company → R ( d , p , ξ ) where ξ has been revealed. → Easy to compute (simulation, Linear Programming). Objective function → f ( d , p ) = E [ R ( d , p , ξ )] . → Remark: f ( d , p ) well defined (for all ( d , p ) ∈ X , R ( d , p , · ) measurable and E [ R ( d , p , ξ )] < ∞ ) and Var [ R ( d , p , ξ )] < ∞ . Laurent DAUDET PhD Defense December 22nd, 2017 15 / 40

  29. Joint scheduling and pricing problem Model The model (2/2) Mathematical model Max ( d , p ) ∈ X f ( d , p ) = E [ R ( d , p , ξ )] where R ( d , p , ξ ) is revenue and X is set of constraints. Laurent DAUDET PhD Defense December 22nd, 2017 16 / 40

  30. Joint scheduling and pricing problem Model The model (2/2) Mathematical model Max ( d , p ) ∈ X f ( d , p ) = E [ R ( d , p , ξ )] where R ( d , p , ξ ) is revenue and X is set of constraints. Compute d and p without knowing ξ ! Laurent DAUDET PhD Defense December 22nd, 2017 16 / 40

  31. Joint scheduling and pricing problem Methods Sample Average Approximation ( ξ 1 , . . . , ξ Ω ) of Ω independent and identically distributed realizations ⇒ ξ ω is not random variable! We approximate objective function f ( d , p ) = E [ R ( d , p , ξ )] by Laurent DAUDET PhD Defense December 22nd, 2017 17 / 40

  32. Joint scheduling and pricing problem Methods Sample Average Approximation ( ξ 1 , . . . , ξ Ω ) of Ω independent and identically distributed realizations ⇒ ξ ω is not random variable! We approximate objective function f ( d , p ) = E [ R ( d , p , ξ )] by Laurent DAUDET PhD Defense December 22nd, 2017 17 / 40

  33. Joint scheduling and pricing problem Methods Sample Average Approximation ( ξ 1 , . . . , ξ Ω ) of Ω independent and identically distributed realizations ⇒ ξ ω is not random variable! We approximate objective function f ( d , p ) = E [ R ( d , p , ξ )] by f Ω ( d , p ) = 1 ˆ � R ( d , p , ξ ω ) Ω ω Laurent DAUDET PhD Defense December 22nd, 2017 17 / 40

  34. Joint scheduling and pricing problem Methods Sample Average Approximation ( ξ 1 , . . . , ξ Ω ) of Ω independent and identically distributed realizations ⇒ ξ ω is not random variable! We approximate objective function f ( d , p ) = E [ R ( d , p , ξ )] by f Ω ( d , p ) = 1 ˆ � R ( d , p , ξ ω ) Ω ω New approximated optimization program Max ( d , p ) ∈ X ˆ f Ω ( d , p ) Laurent DAUDET PhD Defense December 22nd, 2017 17 / 40

  35. Joint scheduling and pricing problem Methods Sample Average Approximation ( ξ 1 , . . . , ξ Ω ) of Ω independent and identically distributed realizations ⇒ ξ ω is not random variable! We approximate objective function f ( d , p ) = E [ R ( d , p , ξ )] by f Ω ( d , p ) = 1 ˆ � R ( d , p , ξ ω ) Ω ω New approximated optimization program Max ( d , p ) ∈ X ˆ f Ω ( d , p ) We denote by → v ∗ = Max ( d , p ) ∈ X f ( d , p ) v Ω = Max ( d , p ) ∈ X ˆ → ˆ f Ω ( d , p ) Laurent DAUDET PhD Defense December 22nd, 2017 17 / 40

  36. Joint scheduling and pricing problem Methods SAA properties Proposition, Shapiro et al., 2009 We have (i) E [ˆ f Ω ( d , p )] = f ( d , p ) , for all ( d , p ) ∈ X , (ii) ˆ f Ω ( d , p ) converges to f ( d , p ) w.p. 1, for all ( d , p ) ∈ X , (iii) E [ˆ v Ω ] ≥ v ∗ , and v Ω converges to v ∗ w.p. 1. (iv) ˆ v Ω ] is an upper bound on v ∗ (iii). E [ˆ → With the Central Limit Theorem, we can compute (1 − α ) -confidence interval. For any solution (¯ p ) ∈ X , f (¯ p ) ≤ v ∗ � � d , ¯ d , ¯ = Max ( d , p ) ∈ X f ( d , p ) ⇒ lower bound on v ∗ . → With (i) and the Central Limit Theorem, we can compute (1 − α ) -confidence interval. Laurent DAUDET PhD Defense December 22nd, 2017 18 / 40

  37. Joint scheduling and pricing problem Methods SAA properties Proposition, Shapiro et al., 2009 We have (i) E [ˆ f Ω ( d , p )] = f ( d , p ) , for all ( d , p ) ∈ X , (ii) ˆ f Ω ( d , p ) converges to f ( d , p ) w.p. 1, for all ( d , p ) ∈ X , (iii) E [ˆ v Ω ] ≥ v ∗ , and v Ω converges to v ∗ w.p. 1. (iv) ˆ v Ω ] is an upper bound on v ∗ (iii). E [ˆ → With the Central Limit Theorem, we can compute (1 − α ) -confidence interval. For any solution (¯ p ) ∈ X , f (¯ p ) ≤ v ∗ � � d , ¯ d , ¯ = Max ( d , p ) ∈ X f ( d , p ) ⇒ lower bound on v ∗ . → With (i) and the Central Limit Theorem, we can compute (1 − α ) -confidence interval. Laurent DAUDET PhD Defense December 22nd, 2017 18 / 40

  38. Joint scheduling and pricing problem Methods SAA properties Proposition, Shapiro et al., 2009 We have (i) E [ˆ f Ω ( d , p )] = f ( d , p ) , for all ( d , p ) ∈ X , (ii) ˆ f Ω ( d , p ) converges to f ( d , p ) w.p. 1, for all ( d , p ) ∈ X , (iii) E [ˆ v Ω ] ≥ v ∗ , and v Ω converges to v ∗ w.p. 1. (iv) ˆ v Ω ] is an upper bound on v ∗ (iii). E [ˆ → With the Central Limit Theorem, we can compute (1 − α ) -confidence interval. For any solution (¯ p ) ∈ X , f (¯ p ) ≤ v ∗ � � d , ¯ d , ¯ = Max ( d , p ) ∈ X f ( d , p ) ⇒ lower bound on v ∗ . → With (i) and the Central Limit Theorem, we can compute (1 − α ) -confidence interval. Laurent DAUDET PhD Defense December 22nd, 2017 18 / 40

  39. Joint scheduling and pricing problem Methods A first heuristic: a sequential heuristic → Try to mimic natural way of scheduling and then fixing prices. 1. Compute departure times d with optimization problem maximizing utilities U ij with prices p = 0 . Laurent DAUDET PhD Defense December 22nd, 2017 19 / 40

  40. Joint scheduling and pricing problem Methods A first heuristic: a sequential heuristic → Try to mimic natural way of scheduling and then fixing prices. 1. Compute departure times d with optimization problem maximizing utilities U ij with prices p = 0 . � Max d ∈ X d E [ U ij ( d , 0 )] i , j Laurent DAUDET PhD Defense December 22nd, 2017 19 / 40

  41. Joint scheduling and pricing problem Methods A first heuristic: a sequential heuristic → Try to mimic natural way of scheduling and then fixing prices. 1. Compute departure times d with optimization problem maximizing utilities U ij with prices p = 0 . � Max d ∈ X d E [ U ij ( d , 0 )] i , j 2. Compute prices p with previous two-stage recourse program (with fixed departure times ˜ d ). Max p ∈ X p ˆ f Ω (˜ d , p ) Laurent DAUDET PhD Defense December 22nd, 2017 19 / 40

  42. Joint scheduling and pricing problem Methods A second heuristic: a Gauss-Seidel heuristic 1. Initialize ( d , p ) to some ( d 0 , p 0 ) . 2. Let m ∈ Z + and J 1 ∪ J 2 ∪ · · · ∪ J m be a partition of { 1 , . . . , S } . 3. Generate sequence of feasible solutions ( d k , p k ) : Laurent DAUDET PhD Defense December 22nd, 2017 20 / 40

  43. Joint scheduling and pricing problem Methods A second heuristic: a Gauss-Seidel heuristic 1. Initialize ( d , p ) to some ( d 0 , p 0 ) . 2. Let m ∈ Z + and J 1 ∪ J 2 ∪ · · · ∪ J m be a partition of { 1 , . . . , S } . 3. Generate sequence of feasible solutions ( d k , p k ) : Laurent DAUDET PhD Defense December 22nd, 2017 20 / 40

  44. Joint scheduling and pricing problem Methods A second heuristic: a Gauss-Seidel heuristic 1. Initialize ( d , p ) to some ( d 0 , p 0 ) . 2. Let m ∈ Z + and J 1 ∪ J 2 ∪ · · · ∪ J m be a partition of { 1 , . . . , S } . 3. Generate sequence of feasible solutions ( d k , p k ) : Laurent DAUDET PhD Defense December 22nd, 2017 20 / 40

  45. Joint scheduling and pricing problem Methods A second heuristic: a Gauss-Seidel heuristic 1. Initialize ( d , p ) to some ( d 0 , p 0 ) . 2. Let m ∈ Z + and J 1 ∪ J 2 ∪ · · · ∪ J m be a partition of { 1 , . . . , S } . 3. Generate sequence of feasible solutions ( d k , p k ) : � ∈ argmax d , p f � d k +1 J 1 , p k +1 � ( d , d k J 2 . . . , d k J m ) , ( p , p k J 2 . . . , p k � J m ) J 1 � ∈ argmax d , p f � d k +1 J ℓ , p k +1 � ( d k +1 J 1 , . . . , d , . . . , d k J m ) , ( p k +1 J 1 , . . . , p , . . . , p k J m ) � J ℓ Laurent DAUDET PhD Defense December 22nd, 2017 20 / 40

  46. Joint scheduling and pricing problem Methods A second heuristic: a Gauss-Seidel heuristic 1. Initialize ( d , p ) to some ( d 0 , p 0 ) . 2. Let m ∈ Z + and J 1 ∪ J 2 ∪ · · · ∪ J m be a partition of { 1 , . . . , S } . 3. Generate sequence of feasible solutions ( d k , p k ) : � ∈ argmax d , p f � d k +1 J 1 , p k +1 � ( d , d k J 2 . . . , d k J m ) , ( p , p k J 2 . . . , p k � J m ) J 1 � ∈ argmax d , p f � d k +1 J 2 , p k +1 � ( d k +1 J 1 , d , d k J 3 , . . . , d k J m ) , ( p k +1 J 1 , p , p k J 3 , . . . , p k J m ) � J 2 Laurent DAUDET PhD Defense December 22nd, 2017 20 / 40

  47. Joint scheduling and pricing problem Methods A second heuristic: a Gauss-Seidel heuristic 1. Initialize ( d , p ) to some ( d 0 , p 0 ) . 2. Let m ∈ Z + and J 1 ∪ J 2 ∪ · · · ∪ J m be a partition of { 1 , . . . , S } . 3. Generate sequence of feasible solutions ( d k , p k ) : � ∈ argmax d , p f � d k +1 J 1 , p k +1 � ( d , d k J 2 . . . , d k J m ) , ( p , p k J 2 . . . , p k � J m ) J 1 � ∈ argmax d , p f � d k +1 J 2 , p k +1 � ( d k +1 J 1 , d , d k J 3 , . . . , d k J m ) , ( p k +1 J 1 , p , p k J 3 , . . . , p k J m ) � J 2 . . . � ∈ argmax d , p f � d k +1 J m , p k +1 � ( d k +1 J 1 , . . ., d k +1 J m − 1 , d ) , ( p k +1 J 1 , . . ., p k +1 � J m − 1 , p ) J m Laurent DAUDET PhD Defense December 22nd, 2017 20 / 40

  48. Joint scheduling and pricing problem Results The results Academic instance: Realistic distribution for preferred departure times. Gumbel distribution for random part of utilities. Instance Lower bounds (20 min) Frontal SAA Seq. heur. G.-S. heur. Q S C 100 3 33 1159.6 ± 0.8 1162.7 ± 1.1 1165.5 ± 1.1 200 5 40 4290.7 ± 2.2 Laurent DAUDET PhD Defense December 22nd, 2017 21 / 40

  49. Joint scheduling and pricing problem Results The results Academic instance: Realistic distribution for preferred departure times. Gumbel distribution for random part of utilities. Instance Lower bounds (20 min) Frontal SAA Seq. heur. G.-S. heur. Q S C 100 3 33 1159.6 ± 0.8 1162.7 ± 1.1 1165.5 ± 1.1 200 5 40 4290.7 ± 2.2 Laurent DAUDET PhD Defense December 22nd, 2017 21 / 40

  50. Joint scheduling and pricing problem Results The results Academic instance: Realistic distribution for preferred departure times. Gumbel distribution for random part of utilities. Instance Lower bounds (20 min) Frontal SAA Seq. heur. G.-S. heur. Q S C 100 3 33 1159.6 ± 0.8 1162.7 ± 1.1 1165.5 ± 1.1 200 5 40 4290.7 ± 2.2 4353.1 ± 2.8 4461.1 ± 2.2 Laurent DAUDET PhD Defense December 22nd, 2017 21 / 40

  51. Joint scheduling and pricing problem Results The results Academic instance: Realistic distribution for preferred departure times. Gumbel distribution for random part of utilities. Instance Lower bounds (20 min) Frontal SAA Seq. heur. G.-S. heur. Q S C 100 3 33 1159.6 ± 0.8 1162.7 ± 1.1 1165.5 ± 1.1 200 5 40 4290.7 ± 2.2 4353.1 ± 2.8 4461.1 ± 2.2 Laurent DAUDET PhD Defense December 22nd, 2017 21 / 40

  52. Joint scheduling and pricing problem Results The results Academic instance: Realistic distribution for preferred departure times. Gumbel distribution for random part of utilities. Instance Lower bounds (20 min) Frontal SAA Seq. heur. G.-S. heur. Q S C 100 3 33 1159.6 ± 0.8 1162.7 ± 1.1 1165.5 ± 1.1 200 5 40 4290.7 ± 2.2 4353.1 ± 2.8 4461.1 ± 2.2 Instance Upper bounds (1 h) Frontal SAA Q S C 100 3 33 1847.4 ± 769.8 200 5 40 5957.1 ± 1339.7 Laurent DAUDET PhD Defense December 22nd, 2017 21 / 40

  53. Joint scheduling and pricing problem Results The results Academic instance: Realistic distribution for preferred departure times. Gumbel distribution for random part of utilities. Instance Lower bounds (20 min) Frontal SAA Seq. heur. G.-S. heur. Q S C 100 3 33 1159.6 ± 0.8 1162.7 ± 1.1 1165.5 ± 1.1 200 5 40 4290.7 ± 2.2 4353.1 ± 2.8 4461.1 ± 2.2 Instance Upper bounds (1 h) Frontal SAA Q S C 100 3 33 1847.4 ± 769.8 200 5 40 5957.1 ± 1339.7 Other heuristic: Lagrangian relaxation ⇒ does not improve Frontal SAA for big instances. Laurent DAUDET PhD Defense December 22nd, 2017 21 / 40

  54. Joint scheduling and pricing problem Results Conclusion Joint scheduling and pricing: higher revenues. Upper bounds: not very precise... Laurent DAUDET PhD Defense December 22nd, 2017 22 / 40

  55. Minimizing the waiting time for a one-way shuttle service Problem General context 1 One-hour schedules maximizing HGV shuttles 2 Joint scheduling and pricing problem 3 Minimizing the waiting time for a one-way shuttle service 4 Laurent DAUDET PhD Defense December 22nd, 2017 22 / 40

  56. Minimizing the waiting time for a one-way shuttle service Problem Why such a problem? Trucks arrive continuously on terminals. Huge waiting lines during peak hours. ⇒ Design schedules to decrease congestion (waiting times). Laurent DAUDET PhD Defense December 22nd, 2017 23 / 40

  57. Minimizing the waiting time for a one-way shuttle service Problem Why such a problem? Trucks arrive continuously on terminals. Huge waiting lines during peak hours. ⇒ Design schedules to decrease congestion (waiting times). Laurent DAUDET PhD Defense December 22nd, 2017 23 / 40

  58. Minimizing the waiting time for a one-way shuttle service Problem Why such a problem? Trucks arrive continuously on terminals. Huge waiting lines during peak hours. ⇒ Design schedules to decrease congestion (waiting times). Laurent DAUDET PhD Defense December 22nd, 2017 23 / 40

  59. Minimizing the waiting time for a one-way shuttle service Problem Why such a problem? Trucks arrive continuously on terminals. Huge waiting lines during peak hours. ⇒ Design schedules to decrease congestion (waiting times). Laurent DAUDET PhD Defense December 22nd, 2017 23 / 40

  60. Minimizing the waiting time for a one-way shuttle service Problem The problem One-way trip. Infinitesimal users arriving continuously (demand known in advance). Company wants to schedule S shuttles of capacity C . Peculiar loading process. ff Laurent DAUDET PhD Defense December 22nd, 2017 24 / 40

  61. Minimizing the waiting time for a one-way shuttle service Problem The problem One-way trip. Infinitesimal users arriving continuously (demand known in advance). Company wants to schedule S shuttles of capacity C . Peculiar loading process. ff Laurent DAUDET PhD Defense December 22nd, 2017 24 / 40

  62. Minimizing the waiting time for a one-way shuttle service Problem The problem One-way trip. Infinitesimal users arriving continuously (demand known in advance). Company wants to schedule S shuttles of capacity C . Peculiar loading process. ff Laurent DAUDET PhD Defense December 22nd, 2017 24 / 40

  63. Minimizing the waiting time for a one-way shuttle service Problem The problem One-way trip. Infinitesimal users arriving continuously (demand known in advance). Company wants to schedule S shuttles of capacity C . Peculiar loading process. ff Laurent DAUDET PhD Defense December 22nd, 2017 24 / 40

  64. Minimizing the waiting time for a one-way shuttle service Problem The loading process Laurent DAUDET PhD Defense December 22nd, 2017 25 / 40

  65. Minimizing the waiting time for a one-way shuttle service Problem The loading process Laurent DAUDET PhD Defense December 22nd, 2017 25 / 40

  66. Minimizing the waiting time for a one-way shuttle service Problem The loading process Laurent DAUDET PhD Defense December 22nd, 2017 25 / 40

  67. Minimizing the waiting time for a one-way shuttle service Problem The problem One-way trip. Infinitesimal users arriving continuously (demand known in advance). Company wants to schedule S shuttles of capacity C . Peculiar loading process. Laurent DAUDET PhD Defense December 22nd, 2017 26 / 40

  68. Minimizing the waiting time for a one-way shuttle service Problem The problem One-way trip. Infinitesimal users arriving continuously (demand known in advance). Company wants to schedule S shuttles of capacity C . Peculiar loading process. Objectives - Minimize maximum waiting time of users � Problem P max - Minimize average waiting time of users � Problem P ave → Waiting time: time between arrival time on terminal and departure of shuttle. Laurent DAUDET PhD Defense December 22nd, 2017 26 / 40

  69. Minimizing the waiting time for a one-way shuttle service Model The model (1/3) Variables d j : departure time j th shuttle. y j : cumulative loads for shuttles 1 to j . Parameters S : number of shuttles. C : capacity. ν : loading rate (loading x users takes a time ν x ). T : time horizon. D : [0 , T ] → R + : cumulative demand known a priori (oracle) → we assume that D ( · ) is upper semicontinuous. Laurent DAUDET PhD Defense December 22nd, 2017 27 / 40

  70. Minimizing the waiting time for a one-way shuttle service Model The model (1/3) Variables d j : departure time j th shuttle. y j : cumulative loads for shuttles 1 to j . Parameters S : number of shuttles. C : capacity. ν : loading rate (loading x users takes a time ν x ). T : time horizon. D : [0 , T ] → R + : cumulative demand known a priori (oracle) → we assume that D ( · ) is upper semicontinuous. Laurent DAUDET PhD Defense December 22nd, 2017 27 / 40

  71. Minimizing the waiting time for a one-way shuttle service Model The model (1/3) Variables d j : departure time j th shuttle. y j : cumulative loads for shuttles 1 to j . Parameters S : number of shuttles. C : capacity. ν : loading rate (loading x users takes a time ν x ). T : time horizon. D : [0 , T ] → R + : cumulative demand known a priori (oracle) → we assume that D ( · ) is upper semicontinuous. Laurent DAUDET PhD Defense December 22nd, 2017 27 / 40

  72. Minimizing the waiting time for a one-way shuttle service Model The model (2/3) Arrival time of user y → function τ ( · ) , pseudo-inverses of D ( · ) : τ ( y ) = inf { t ∈ [0 , T ]: D ( t ) ≥ y } . Objective function � y j 1 g ave ( d , y ) = � ( d j − τ ( y )) dy D ( T ) y j − 1 j Laurent DAUDET PhD Defense December 22nd, 2017 28 / 40

  73. Minimizing the waiting time for a one-way shuttle service Model The model (2/3) Arrival time of user y → function τ ( · ) , pseudo-inverses of D ( · ) : τ ( y ) = inf { t ∈ [0 , T ]: D ( t ) ≥ y } . Objective function � y j 1 g ave ( d , y ) = � ( d j − τ ( y )) dy D ( T ) y j − 1 j Laurent DAUDET PhD Defense December 22nd, 2017 28 / 40

  74. Minimizing the waiting time for a one-way shuttle service Model The model (2/3) Arrival time of user y → function τ ( · ) , pseudo-inverses of D ( · ) : τ ( y ) = inf { t ∈ [0 , T ]: D ( t ) ≥ y } . Objective function � y j 1 g ave ( d , y ) = � ( d j − τ ( y )) dy D ( T ) y j − 1 j Similar function g max ( d , y ) for maximum waiting time. Laurent DAUDET PhD Defense December 22nd, 2017 28 / 40

  75. Minimizing the waiting time for a one-way shuttle service Model The model (3/3) Min d , y g ( d , y ) s.t. y j − y j − 1 ≤ C y j − 1 ≤ y j ≤ d j − 1 d j y S = D ( T ) ≥ τ ( y j ) + ν ( y j − y j − 1 ) d j y 0 = 0 . Laurent DAUDET PhD Defense December 22nd, 2017 29 / 40

  76. Minimizing the waiting time for a one-way shuttle service Piecewise constant demand When demand is a step function Theorem Assume that D ( · ) is a step function defined with K discontinuities, and that ν = 0 . There is an algorithm computing an optimal solution of P ave in O ( K 2 S ) . Laurent DAUDET PhD Defense December 22nd, 2017 30 / 40

  77. Minimizing the waiting time for a one-way shuttle service Piecewise constant demand Precision on ν = 0 Laurent DAUDET PhD Defense December 22nd, 2017 31 / 40

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