Algorithms for train scheduling on a single line Laurent DAUDET 1 PhD - - PowerPoint PPT Presentation

Algorithms for train scheduling on a single line

Laurent DAUDET1

PhD advisor: Frédéric MEUNIER1

1CERMICS, Centre d’Enseignement et de Recherche en Mathématiques et Calcul Scientifique,

ENPC December 22nd, 2017

Laurent DAUDET PhD Defense December 22nd, 2017 1 / 40

General context Company

The context

→ Scientific chair between

École Nationale des Ponts et Chaussées Eurotunnel

Laurent DAUDET PhD Defense December 22nd, 2017 2 / 40

General context Company

The context

→ Scientific chair between

École Nationale des Ponts et Chaussées Eurotunnel

Questions? How to increase the global capacity with current means? How to improve the quality of service?

Laurent DAUDET PhD Defense December 22nd, 2017 2 / 40

General context Company

The tunnel under the Channel

Line length: 50 km From Coquelles (France) to Folkestone (England) A tunnel for each direction (A) and a service tunnel (B) High-Speed-Trains (Eurostar) 160 km/h Freight trains 100-120 km/h Passenger shuttles (PAX) 140 km/h Freight shuttles (HGV) 140 km/h

Laurent DAUDET PhD Defense December 22nd, 2017 3 / 40

General context Company

The tunnel under the Channel

Line length: 50 km From Coquelles (France) to Folkestone (England) A tunnel for each direction (A) and a service tunnel (B) High-Speed-Trains (Eurostar) 160 km/h Freight trains 100-120 km/h Passenger shuttles (PAX) 140 km/h Freight shuttles (HGV) 140 km/h

Laurent DAUDET PhD Defense December 22nd, 2017 3 / 40

General context Company

The tunnel under the Channel

Line length: 50 km From Coquelles (France) to Folkestone (England) A tunnel for each direction (A) and a service tunnel (B) High-Speed-Trains (Eurostar) 160 km/h Freight trains 100-120 km/h Passenger shuttles (PAX) 140 km/h Freight shuttles (HGV) 140 km/h

Laurent DAUDET PhD Defense December 22nd, 2017 3 / 40

General context Company

1

General context

2

One-hour schedules maximizing HGV shuttles

3

Joint scheduling and pricing problem

4

Minimizing the waiting time for a one-way shuttle service

Laurent DAUDET PhD Defense December 22nd, 2017 4 / 40

One-hour schedules maximizing HGV shuttles Problem

1

General context

2

One-hour schedules maximizing HGV shuttles

3

Joint scheduling and pricing problem

4

Minimizing the waiting time for a one-way shuttle service

Laurent DAUDET PhD Defense December 22nd, 2017 4 / 40

One-hour schedules maximizing HGV shuttles Problem

What are the goals?

Confirm optimality of current schedules. Compute schedules with new instances. “Price” the constraints for future investments and negotiations.

Laurent DAUDET PhD Defense December 22nd, 2017 5 / 40

One-hour schedules maximizing HGV shuttles Problem

What are the goals?

Confirm optimality of current schedules. Compute schedules with new instances. “Price” the constraints for future investments and negotiations.

Laurent DAUDET PhD Defense December 22nd, 2017 5 / 40

One-hour schedules maximizing HGV shuttles Problem

What are the goals?

Confirm optimality of current schedules. Compute schedules with new instances. “Price” the constraints for future investments and negotiations.

Laurent DAUDET PhD Defense December 22nd, 2017 5 / 40

One-hour schedules maximizing HGV shuttles Problem

What are the goals?

Confirm optimality of current schedules. Compute schedules with new instances. “Price” the constraints for future investments and negotiations.

Laurent DAUDET PhD Defense December 22nd, 2017 5 / 40

One-hour schedules maximizing HGV shuttles Problem

The problem

Objective Compute one-hour schedule with maximum number of HGV shuttles. Constraints Fixed number of Eurostars, freight trains, and PAX shuttles. Security. Other constraints. Output d =

• dEur

j , dFr j , dPAX j

, dHGV

j

• : scheduled departure times of all trains.

→ dA

j : jth departure time of train of type A.

Laurent DAUDET PhD Defense December 22nd, 2017 6 / 40

One-hour schedules maximizing HGV shuttles Problem

The problem

Objective Compute one-hour schedule with maximum number of HGV shuttles. Constraints Fixed number of Eurostars, freight trains, and PAX shuttles. Security. Other constraints. Output d =

• dEur

j , dFr j , dPAX j

, dHGV

j

• : scheduled departure times of all trains.

→ dA

j : jth departure time of train of type A.

Laurent DAUDET PhD Defense December 22nd, 2017 6 / 40

One-hour schedules maximizing HGV shuttles Problem

The problem

Objective Compute one-hour schedule with maximum number of HGV shuttles. Constraints Fixed number of Eurostars, freight trains, and PAX shuttles. Security. Other constraints. Output d =

• dEur

j , dFr j , dPAX j

, dHGV

j

• : scheduled departure times of all trains.

→ dA

j : jth departure time of train of type A.

Laurent DAUDET PhD Defense December 22nd, 2017 6 / 40

One-hour schedules maximizing HGV shuttles Problem

Some other constraints

Commercial agreements: Eurostars “equally” distributed in the period. → Eurostars grouped by pairs with departure times at d and d + 30 min. Loading platforms: impossible to load three HGV or PAX shuttles at the same time. → at most 2 HGV shuttles in any 12-minute time-window. → at most 2 PAX shuttles in any 12-minute time-window. Discretization: departure times on full minutes. ...

Laurent DAUDET PhD Defense December 22nd, 2017 7 / 40

One-hour schedules maximizing HGV shuttles Problem

Some other constraints

Commercial agreements: Eurostars “equally” distributed in the period. → Eurostars grouped by pairs with departure times at d and d + 30 min. Loading platforms: impossible to load three HGV or PAX shuttles at the same time. → at most 2 HGV shuttles in any 12-minute time-window. → at most 2 PAX shuttles in any 12-minute time-window. Discretization: departure times on full minutes. ...

Laurent DAUDET PhD Defense December 22nd, 2017 7 / 40

One-hour schedules maximizing HGV shuttles Problem

Some other constraints

Commercial agreements: Eurostars “equally” distributed in the period. → Eurostars grouped by pairs with departure times at d and d + 30 min. Loading platforms: impossible to load three HGV or PAX shuttles at the same time. → at most 2 HGV shuttles in any 12-minute time-window. → at most 2 PAX shuttles in any 12-minute time-window. Discretization: departure times on full minutes. ...

Laurent DAUDET PhD Defense December 22nd, 2017 7 / 40

One-hour schedules maximizing HGV shuttles Problem

Some other constraints

Commercial agreements: Eurostars “equally” distributed in the period. → Eurostars grouped by pairs with departure times at d and d + 30 min. Loading platforms: impossible to load three HGV or PAX shuttles at the same time. → at most 2 HGV shuttles in any 12-minute time-window. → at most 2 PAX shuttles in any 12-minute time-window. Discretization: departure times on full minutes. ...

Laurent DAUDET PhD Defense December 22nd, 2017 7 / 40

One-hour schedules maximizing HGV shuttles Model

The model

Variables d =

• dEur

j , dFr j , dPAX j

, dHGV

j

• : scheduled departure times of all trains.

→ dA

j : jth departure time of train of type A.

n: number of scheduled HGV shuttles. Mathematical model 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

One-hour schedules maximizing HGV shuttles Model

The model

Variables d =

• dEur

j , dFr j , dPAX j

, dHGV

j

• : scheduled departure times of all trains.

→ dA

j : jth departure time of train of type A.

n: number of scheduled HGV shuttles. Mathematical model Max( 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

One-hour schedules maximizing HGV shuttles Model

The model

Variables d =

• dEur

j , dFr j , dPAX j

, dHGV

j

• : scheduled departure times of all trains.

→ dA

j : jth departure time of train of type 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

One-hour schedules maximizing HGV shuttles Model

The model

Variables d =

• dEur

j , dFr j , dPAX j

, dHGV

j

• : scheduled departure times of all trains.

→ dA

j : jth departure time of train of type 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

One-hour schedules maximizing HGV shuttles Model

The model

Variables d =

• dEur

j , dFr j , dPAX j

, dHGV

j

• : scheduled departure times of all trains.

→ dA

j : jth departure time of train of type 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

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

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

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

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

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

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

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

Joint scheduling and pricing problem Problem

1

General context

2

One-hour schedules maximizing HGV shuttles

3

Joint scheduling and pricing problem

4

Minimizing the waiting time for a one-way shuttle service

Laurent DAUDET PhD Defense December 22nd, 2017 11 / 40

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

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

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

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

Joint scheduling and pricing problem Problem

The problem (1/2)

One-way trip.

Laurent DAUDET PhD Defense December 22nd, 2017 13 / 40

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

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

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

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

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

Joint scheduling and pricing problem Problem

The problem (2/2)

Each customer i has a preferred departure time: random variable χi belongs to economic class bi (e.g. business, tourist, low-cost, ...) “value of time” vb for economic class b

Laurent DAUDET PhD Defense December 22nd, 2017 14 / 40

Joint scheduling and pricing problem Problem

The problem (2/2)

Each customer i has a preferred departure time: random variable χi belongs to economic class bi (e.g. business, tourist, low-cost, ...) “value of time” vb for economic class b

Laurent DAUDET PhD Defense December 22nd, 2017 14 / 40

Joint scheduling and pricing problem Problem

The problem (2/2)

Each customer i has a preferred departure time: random variable χi belongs to economic class bi (e.g. business, tourist, low-cost, ...) “value of time” vb for economic class b

Laurent DAUDET PhD Defense December 22nd, 2017 14 / 40

Joint scheduling and pricing problem Problem

The problem (2/2)

Each customer i has a preferred departure time: random variable χi belongs to economic class bi (e.g. business, tourist, ...) “value of time” vb for economic class b We assume v1 ≤ v2 ≤ · · · Customers of class 1 make their choice first, then 2, ...

Laurent DAUDET PhD Defense December 22nd, 2017 14 / 40

Joint scheduling and pricing problem Problem

The problem (2/2)

Each customer i has a preferred departure time: random variable χi belongs to economic class bi (e.g. business, tourist, ...) “value of time” vb for economic class b We assume v1 ≤ v2 ≤ · · · Customers of class 1 make their choice first, then 2, ... Discrete choice model Each customer i and product j (departure dj at price pj) → Random utility Uij(d, p) representing satisfaction.

Laurent DAUDET PhD Defense December 22nd, 2017 14 / 40

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

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

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

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

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

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

Joint scheduling and pricing problem Methods

Sample Average Approximation

(ξ1, . . . , ξΩ) of Ω independent and identically distributed realizations ⇒ ξω 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

Joint scheduling and pricing problem Methods

Sample Average Approximation

(ξ1, . . . , ξΩ) of Ω independent and identically distributed realizations ⇒ ξω 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

Joint scheduling and pricing problem Methods

Sample Average Approximation

(ξ1, . . . , ξΩ) of Ω independent and identically distributed realizations ⇒ ξω is not random variable! We approximate objective function f (d, p) = E [R(d, p, ξ)] by ˆ fΩ(d, p) = 1 Ω

• ω

R(d, p, ξω)

Laurent DAUDET PhD Defense December 22nd, 2017 17 / 40

Joint scheduling and pricing problem Methods

Sample Average Approximation

(ξ1, . . . , ξΩ) of Ω independent and identically distributed realizations ⇒ ξω is not random variable! We approximate objective function f (d, p) = E [R(d, p, ξ)] by ˆ fΩ(d, p) = 1 Ω

• ω

R(d, p, ξω) New approximated optimization program Max(d,p)∈X ˆ fΩ(d, p)

Laurent DAUDET PhD Defense December 22nd, 2017 17 / 40

Joint scheduling and pricing problem Methods

Sample Average Approximation

(ξ1, . . . , ξΩ) of Ω independent and identically distributed realizations ⇒ ξω is not random variable! We approximate objective function f (d, p) = E [R(d, p, ξ)] by ˆ fΩ(d, p) = 1 Ω

• ω

R(d, p, ξω) New approximated optimization program Max(d,p)∈X ˆ fΩ(d, p) We denote by → v∗ = Max(d,p)∈X f (d, p) → ˆ vΩ = Max(d,p)∈X ˆ fΩ(d, p)

Laurent DAUDET PhD Defense December 22nd, 2017 17 / 40

Joint scheduling and pricing problem Methods

SAA properties

Proposition, Shapiro et al., 2009 We have (i) E[ˆ fΩ(d, p)] = f (d, p), for all (d, p) ∈ X, (ii) ˆ fΩ(d, p) converges to f (d, p) w.p. 1, for all (d, p) ∈ X, (iii) E[ˆ vΩ] ≥ v∗, and (iv) ˆ vΩ converges to v∗ w.p. 1. E[ˆ vΩ] is an upper bound on v∗ (iii). → With the Central Limit Theorem, we can compute (1 − α)-confidence interval. For any solution (¯ d, ¯ p) ∈ X, f (¯ d, ¯ p) ≤ v∗ = 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

Joint scheduling and pricing problem Methods

SAA properties

Proposition, Shapiro et al., 2009 We have (i) E[ˆ fΩ(d, p)] = f (d, p), for all (d, p) ∈ X, (ii) ˆ fΩ(d, p) converges to f (d, p) w.p. 1, for all (d, p) ∈ X, (iii) E[ˆ vΩ] ≥ v∗, and (iv) ˆ vΩ converges to v∗ w.p. 1. E[ˆ vΩ] is an upper bound on v∗ (iii). → With the Central Limit Theorem, we can compute (1 − α)-confidence interval. For any solution (¯ d, ¯ p) ∈ X, f (¯ d, ¯ p) ≤ v∗ = 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

Joint scheduling and pricing problem Methods

SAA properties

Proposition, Shapiro et al., 2009 We have (i) E[ˆ fΩ(d, p)] = f (d, p), for all (d, p) ∈ X, (ii) ˆ fΩ(d, p) converges to f (d, p) w.p. 1, for all (d, p) ∈ X, (iii) E[ˆ vΩ] ≥ v∗, and (iv) ˆ vΩ converges to v∗ w.p. 1. E[ˆ vΩ] is an upper bound on v∗ (iii). → With the Central Limit Theorem, we can compute (1 − α)-confidence interval. For any solution (¯ d, ¯ p) ∈ X, f (¯ d, ¯ p) ≤ v∗ = 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

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 Uij with prices p = 0.

Laurent DAUDET PhD Defense December 22nd, 2017 19 / 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 Uij with prices p = 0. Maxd∈Xd

• i,j

E [Uij(d, 0)]

Laurent DAUDET PhD Defense December 22nd, 2017 19 / 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 Uij with prices p = 0. Maxd∈Xd

• i,j

E [Uij(d, 0)]

• 2. Compute prices p with previous two-stage recourse program (with fixed

departure times ˜ d). Maxp∈Xp ˆ fΩ(˜ d, p)

Laurent DAUDET PhD Defense December 22nd, 2017 19 / 40

Joint scheduling and pricing problem Methods

A second heuristic: a Gauss-Seidel heuristic

• 1. Initialize (d, p) to some (d0, p0).
• 2. Let m ∈ Z+ and J1 ∪ J2 ∪ · · · ∪ Jm be a partition of {1, . . . , S}.
• 3. Generate sequence of feasible solutions (dk, pk):

Laurent DAUDET PhD Defense December 22nd, 2017 20 / 40

Joint scheduling and pricing problem Methods

A second heuristic: a Gauss-Seidel heuristic

• 1. Initialize (d, p) to some (d0, p0).
• 2. Let m ∈ Z+ and J1 ∪ J2 ∪ · · · ∪ Jm be a partition of {1, . . . , S}.
• 3. Generate sequence of feasible solutions (dk, pk):

Laurent DAUDET PhD Defense December 22nd, 2017 20 / 40

Joint scheduling and pricing problem Methods

A second heuristic: a Gauss-Seidel heuristic

• 1. Initialize (d, p) to some (d0, p0).
• 2. Let m ∈ Z+ and J1 ∪ J2 ∪ · · · ∪ Jm be a partition of {1, . . . , S}.
• 3. Generate sequence of feasible solutions (dk, pk):

Laurent DAUDET PhD Defense December 22nd, 2017 20 / 40

Joint scheduling and pricing problem Methods

A second heuristic: a Gauss-Seidel heuristic

• 1. Initialize (d, p) to some (d0, p0).
• 2. Let m ∈ Z+ and J1 ∪ J2 ∪ · · · ∪ Jm be a partition of {1, . . . , S}.
• 3. Generate sequence of feasible solutions (dk, pk):

dk+1

J1 , pk+1 J1

∈ argmaxd,p f (d, dk

J2 . . . , dk Jm), (p, pk J2 . . . , pk Jm)

• dk+1

Jℓ , pk+1 Jℓ

∈ argmaxd,p f (dk+1

J1 , . . . , d, . . . , dk Jm), (pk+1 J1 , . . . , p, . . . , pk Jm)

• Laurent DAUDET

PhD Defense December 22nd, 2017 20 / 40

Joint scheduling and pricing problem Methods

A second heuristic: a Gauss-Seidel heuristic

• 1. Initialize (d, p) to some (d0, p0).
• 2. Let m ∈ Z+ and J1 ∪ J2 ∪ · · · ∪ Jm be a partition of {1, . . . , S}.
• 3. Generate sequence of feasible solutions (dk, pk):

dk+1

J1 , pk+1 J1

∈ argmaxd,p f (d, dk

J2 . . . , dk Jm), (p, pk J2 . . . , pk Jm)

• dk+1

J2 , pk+1 J2

∈ argmaxd,p f (dk+1

J1 , d, dk J3, . . . , dk Jm), (pk+1 J1 , p, pk J3, . . . , pk Jm)

• Laurent DAUDET

PhD Defense December 22nd, 2017 20 / 40

Joint scheduling and pricing problem Methods

A second heuristic: a Gauss-Seidel heuristic

• 1. Initialize (d, p) to some (d0, p0).
• 2. Let m ∈ Z+ and J1 ∪ J2 ∪ · · · ∪ Jm be a partition of {1, . . . , S}.
• 3. Generate sequence of feasible solutions (dk, pk):

dk+1

J1 , pk+1 J1

∈ argmaxd,p f (d, dk

J2 . . . , dk Jm), (p, pk J2 . . . , pk Jm)

• dk+1

J2 , pk+1 J2

∈ argmaxd,p f (dk+1

J1 , d, dk J3, . . . , dk Jm), (pk+1 J1 , p, pk J3, . . . , pk Jm)

• .

. .

dk+1

Jm , pk+1 Jm

∈ argmaxd,p f (dk+1

J1 , . . ., dk+1 Jm−1, d), (pk+1 J1 , . . ., pk+1 Jm−1, p)

• Laurent DAUDET

PhD Defense December 22nd, 2017 20 / 40

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) Q S C Frontal SAA

• Seq. heur.

G.-S. heur. 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

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) Q S C Frontal SAA

• Seq. heur.

G.-S. heur. 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

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) Q S C Frontal SAA

• Seq. heur.

G.-S. heur. 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

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) Q S C Frontal SAA

• Seq. heur.

G.-S. heur. 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

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) Q S C Frontal SAA

• Seq. heur.

G.-S. heur. 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) Q S C Frontal SAA 100 3 33 1847.4± 769.8 200 5 40 5957.1± 1339.7

Laurent DAUDET PhD Defense December 22nd, 2017 21 / 40

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) Q S C Frontal SAA

• Seq. heur.

G.-S. heur. 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) Q S C Frontal SAA 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

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

Minimizing the waiting time for a one-way shuttle service Problem

1

General context

2

One-hour schedules maximizing HGV shuttles

3

Joint scheduling and pricing problem

4

Minimizing the waiting time for a one-way shuttle service

Laurent DAUDET PhD Defense December 22nd, 2017 22 / 40

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

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

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

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

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

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

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

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

Minimizing the waiting time for a one-way shuttle service Problem

The loading process

Laurent DAUDET PhD Defense December 22nd, 2017 25 / 40

Minimizing the waiting time for a one-way shuttle service Problem

The loading process

Laurent DAUDET PhD Defense December 22nd, 2017 25 / 40

Minimizing the waiting time for a one-way shuttle service Problem

The loading process

Laurent DAUDET PhD Defense December 22nd, 2017 25 / 40

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

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 Pmax

• Minimize average waiting time of users

Problem Pave → Waiting time: time between arrival time on terminal and departure of shuttle.

Laurent DAUDET PhD Defense December 22nd, 2017 26 / 40

Minimizing the waiting time for a one-way shuttle service Model

The model (1/3)

Variables dj: departure time jth shuttle. yj: cumulative loads for shuttles 1 to j. Parameters S: number of shuttles. C: capacity. ν: loading rate (loading x users takes a time νx). 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

Minimizing the waiting time for a one-way shuttle service Model

The model (1/3)

Variables dj: departure time jth shuttle. yj: cumulative loads for shuttles 1 to j. Parameters S: number of shuttles. C: capacity. ν: loading rate (loading x users takes a time νx). 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

Minimizing the waiting time for a one-way shuttle service Model

The model (1/3)

Variables dj: departure time jth shuttle. yj: cumulative loads for shuttles 1 to j. Parameters S: number of shuttles. C: capacity. ν: loading rate (loading x users takes a time νx). 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

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 gave(d, y) = 1 D(T)

• j

yj

yj−1

(dj − τ(y)) dy

Laurent DAUDET PhD Defense December 22nd, 2017 28 / 40

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 gave(d, y) = 1 D(T)

• j

yj

yj−1

(dj − τ(y)) dy

Laurent DAUDET PhD Defense December 22nd, 2017 28 / 40

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 gave(d, y) = 1 D(T)

• j

yj

yj−1

(dj − τ(y)) dy Similar function gmax(d, y) for maximum waiting time.

Laurent DAUDET PhD Defense December 22nd, 2017 28 / 40

Minimizing the waiting time for a one-way shuttle service Model

The model (3/3)

Mind,y g(d, y) s.t. yj − yj−1 ≤ C yj−1 ≤ yj dj−1 ≤ dj yS = D(T) dj ≥ τ(yj) + ν(yj − yj−1) y0 = 0.

Laurent DAUDET PhD Defense December 22nd, 2017 29 / 40

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 Pave in O(K 2S).

Laurent DAUDET PhD Defense December 22nd, 2017 30 / 40

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

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

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

Minimizing the waiting time for a one-way shuttle service Piecewise constant demand

An algorithm for Pave and C = ∞

Shortest path in S arcs minimizing sum on arcs in directed acircuitic graph with K + 1 vertices ⇒ complexity O(K 2S). C < ∞, idem in graph with O(K) vertices.

Laurent DAUDET PhD Defense December 22nd, 2017 32 / 40

Minimizing the waiting time for a one-way shuttle service Piecewise constant demand

An algorithm for Pave and C = ∞

(d2,y2) (d1,y1) y2 y1

Shortest path in S arcs minimizing sum on arcs in directed acircuitic graph with K + 1 vertices ⇒ complexity O(K 2S). C < ∞, idem in graph with O(K) vertices.

Laurent DAUDET PhD Defense December 22nd, 2017 32 / 40

Minimizing the waiting time for a one-way shuttle service Piecewise constant demand

An algorithm for Pave and C = ∞

(d2,y2) (d1,y1) y2 y1

Shortest path in S arcs minimizing sum on arcs in directed acircuitic graph with K + 1 vertices ⇒ complexity O(K 2S). C < ∞, idem in graph with O(K) vertices.

Laurent DAUDET PhD Defense December 22nd, 2017 32 / 40

Minimizing the waiting time for a one-way shuttle service Piecewise constant demand

An algorithm for Pave and C = ∞

(d2,y2) (d1,y1) y2 y1

Shortest path in S arcs minimizing sum on arcs in directed acircuitic graph with K + 1 vertices ⇒ complexity O(K 2S). C < ∞, idem in graph with O(K) vertices.

Laurent DAUDET PhD Defense December 22nd, 2017 32 / 40

Minimizing the waiting time for a one-way shuttle service Piecewise constant demand

An algorithm for Pave and C = ∞

S 4 3 2 1

Shortest path in S arcs minimizing sum on arcs in directed acircuitic graph with K + 1 vertices ⇒ complexity O(K 2S). C < ∞, idem in graph with O(K) vertices.

Laurent DAUDET PhD Defense December 22nd, 2017 32 / 40

Minimizing the waiting time for a one-way shuttle service Piecewise constant demand

An algorithm for Pave and C = ∞

Shortest path in S arcs minimizing sum on arcs in directed acircuitic graph with K + 1 vertices ⇒ complexity O(K 2S). C < ∞, idem in graph with O(K) vertices.

Laurent DAUDET PhD Defense December 22nd, 2017 32 / 40

Minimizing the waiting time for a one-way shuttle service Piecewise constant demand

An algorithm for Pave and C = ∞

Shortest path in S arcs minimizing sum on arcs in directed acircuitic graph with K + 1 vertices ⇒ complexity O(K 2S). C < ∞, idem in graph with O(K) vertices.

Laurent DAUDET PhD Defense December 22nd, 2017 32 / 40

Minimizing the waiting time for a one-way shuttle service Piecewise constant demand

An algorithm for Pave and C = ∞

Shortest path in S arcs minimizing sum on arcs in directed acircuitic graph with K + 1 vertices ⇒ complexity O(K 2S). C < ∞, idem in graph with O(K) vertices.

Laurent DAUDET PhD Defense December 22nd, 2017 32 / 40

Minimizing the waiting time for a one-way shuttle service Approximation scheme for Pave

An approximation scheme for Pave

Theorem Suppose that D( · ) admits right derivatives everywhere (denoted D′

+(t))

and inft∈[0,T) D′

+(t) is positive. Then, for any positive integer M, a feasible

solution of value SOL ≤ OPT + O

• S2

M

• can be computed in O

SM 3.

Laurent DAUDET PhD Defense December 22nd, 2017 33 / 40

Minimizing the waiting time for a one-way shuttle service Approximation scheme for Pave

Some elements of proof (1/2)

Laurent DAUDET PhD Defense December 22nd, 2017 34 / 40

Minimizing the waiting time for a one-way shuttle service Approximation scheme for Pave

Some elements of proof (1/2)

Lemma There exists a collection of problems (Pη), with parameter η, providing: (i) lower bound LBη of OPT. (ii) upper bound UBη of OPT. (iii) lim

η→0 UBη − LBη = 0.

Laurent DAUDET PhD Defense December 22nd, 2017 34 / 40

Minimizing the waiting time for a one-way shuttle service Approximation scheme for Pave

Some elements of proof (1/2)

Lemma There exists a collection of problems (Pη), with parameter η, providing: (i) lower bound LBη of OPT. (ii) upper bound UBη of OPT. (iii) lim

η→0 UBη − LBη = 0.

Laurent DAUDET PhD Defense December 22nd, 2017 34 / 40

Minimizing the waiting time for a one-way shuttle service Approximation scheme for Pave

Some elements of proof (1/2)

Lemma There exists a collection of problems (Pη), with parameter η, providing: (i) lower bound LBη of OPT. (ii) upper bound UBη of OPT. (iii) lim

η→0 UBη − LBη = 0.

Laurent DAUDET PhD Defense December 22nd, 2017 34 / 40

Minimizing the waiting time for a one-way shuttle service Approximation scheme for Pave

Some elements of proof (1/2)

Lemma There exists a collection of problems (Pη), with parameter η, providing: (i) lower bound LBη of OPT. (ii) upper bound UBη of OPT. (iii) lim

η→0 UBη − LBη = 0.

Laurent DAUDET PhD Defense December 22nd, 2017 34 / 40

Minimizing the waiting time for a one-way shuttle service Approximation scheme for Pave

Some elements of proof (1/2)

Lemma There exists a collection of problems (Pη), with parameter η, providing: (i) lower bound LBη of OPT. (ii) upper bound UBη of OPT. (iii) lim

η→0 UBη − LBη = 0.

→ η depends on M. → Problems (Pη): Demand is discretized with step η ⇒ Shortest path in directed acircuitic graph.

Laurent DAUDET PhD Defense December 22nd, 2017 34 / 40

Minimizing the waiting time for a one-way shuttle service Approximation scheme for Pave

Elements of proof (2/2)

Laurent DAUDET PhD Defense December 22nd, 2017 35 / 40

Minimizing the waiting time for a one-way shuttle service Approximation scheme for Pave

Existence of optimal solution of Pave?

τ( · ) lower semicontinuous → set of constraints is closed. ⇒ set of constraints is compact! gave(d, y) continuous.

Laurent DAUDET PhD Defense December 22nd, 2017 36 / 40

Minimizing the waiting time for a one-way shuttle service Approximation scheme for Pave

Existence of optimal solution of Pave?

Mind,y gave(d, y) = 1 D(T)

• j

yj

yj−1

(dj − τ(y)) dy s.t. yj − yj−1 ≤ C yj−1 ≤ yj dj−1 ≤ dj yS = D(T) dj ≥ τ(yj) + ν(yj − yj−1) y0 = 0. dj ≤ T + νC τ( · ) lower semicontinuous → set of constraints is closed. ⇒ set of constraints is compact! gave(d, y) continuous.

Laurent DAUDET PhD Defense December 22nd, 2017 36 / 40

Minimizing the waiting time for a one-way shuttle service Approximation scheme for Pave

Existence of optimal solution of Pave?

Mind,y gave(d, y) = 1 D(T)

• j

yj

yj−1

(dj − τ(y)) dy s.t. yj − yj−1 ≤ C yj−1 ≤ yj dj−1 ≤ dj yS = D(T) dj ≥ τ(yj) + ν(yj − yj−1) y0 = dj ≤ T + νC. τ( · ) lower semicontinuous → set of constraints is closed. ⇒ set of constraints is compact! gave(d, y) continuous.

Laurent DAUDET PhD Defense December 22nd, 2017 36 / 40

Minimizing the waiting time for a one-way shuttle service Approximation scheme for Pave

Existence of optimal solution of Pave?

Mind,y gave(d, y) = 1 D(T)

• j

yj

yj−1

(dj − τ(y)) dy s.t. yj − yj−1 ≤ C yj−1 ≤ yj dj−1 ≤ dj yS = D(T) dj ≥ τ(yj) + ν(yj − yj−1) y0 = dj ≤ T + νC. τ( · ) lower semicontinuous → set of constraints is closed. ⇒ set of constraints is compact! gave(d, y) continuous.

Laurent DAUDET PhD Defense December 22nd, 2017 36 / 40

Minimizing the waiting time for a one-way shuttle service Approximation scheme for Pave

Existence of optimal solution of Pave?

Mind,y gave(d, y) = 1 D(T)

• j

yj

yj−1

(dj − τ(y)) dy s.t. yj − yj−1 ≤ C yj−1 ≤ yj dj−1 ≤ dj yS = D(T) dj ≥ τ(yj) + ν(yj − yj−1) y0 = dj ≤ T + νC. τ( · ) lower semicontinuous → set of constraints is closed. ⇒ set of constraints is compact! gave(d, y) continuous.

Laurent DAUDET PhD Defense December 22nd, 2017 36 / 40

Minimizing the waiting time for a one-way shuttle service Approximation scheme for Pave

Existence of optimal solution of Pave?

Mind,y gave(d, y) = 1 D(T)

• j

yj

yj−1

(dj − τ(y)) dy s.t. yj − yj−1 ≤ C yj−1 ≤ yj dj−1 ≤ dj yS = D(T) dj ≥ τ(yj) + ν(yj − yj−1) y0 = dj ≤ T + νC. τ( · ) lower semicontinuous → set of constraints is closed. ⇒ set of constraints is compact! gave(d, y) continuous.

Laurent DAUDET PhD Defense December 22nd, 2017 36 / 40

Minimizing the waiting time for a one-way shuttle service Results

Table of results

Pmax Pave

Exact algorithm O(K 2S) Approx. algorithm O

• S log S

ε

• Approx.

algorithm O

SM 3

Laurent DAUDET PhD Defense December 22nd, 2017 37 / 40

Minimizing the waiting time for a one-way shuttle service Results

Table of results

Pmax Pave

Exact algorithm O(K 2S) Exact algorithm O(K 2S) Approx. algorithm O

• S log S

ε

• Approx.

algorithm O

SM 3

Laurent DAUDET PhD Defense December 22nd, 2017 37 / 40

Minimizing the waiting time for a one-way shuttle service Results

Table of results

Pmax Pave

Exact algorithm O(K 2S) Exact algorithm O(K 2S) Approx. algorithm O

• S log S

ε

• Approx.

algorithm O

SM 3

Laurent DAUDET PhD Defense December 22nd, 2017 37 / 40

Minimizing the waiting time for a one-way shuttle service Results

Table of results

Pmax Pave Pmax

return

Pave

return

Exact algorithm O(K 2S) Exact algorithm O(K 2S) Approx. algorithm O

• S log S

ε

• Approx.

algorithm O

SM 3

Laurent DAUDET PhD Defense December 22nd, 2017 37 / 40

Minimizing the waiting time for a one-way shuttle service Results

Table of results

Pmax Pave Pmax

return

Pave

return

Exact algorithm O(K 2S) Exact algorithm O(K 2S) Approx. algorithm O

• S log S

ε

• Approx.

algorithm O

SM 3

Approx. algorithm O

• β3SM 2S+1

Laurent DAUDET PhD Defense December 22nd, 2017 37 / 40

Minimizing the waiting time for a one-way shuttle service Results

Table of results

Pmax Pave Pmax

return

Pave

return

Exact algorithm O(K 2S) Exact algorithm O(K 2S)

? ?

Approx. algorithm O

• S log S

ε

• Approx.

algorithm O

SM 3

Approx. algorithm O

• β3SM 2S+1

?

Laurent DAUDET PhD Defense December 22nd, 2017 37 / 40

Minimizing the waiting time for a one-way shuttle service Results

Table of results

Pmax Pave Pmax

return

Pave

return

Exact algorithm O(K 2S) Exact algorithm O(K 2S)

? ?

Approx. algorithm O

• S log S

ε

• Approx.

algorithm O

SM 3

Approx. algorithm O

• β3SM 2S+1

?

Closed-form expression O(1) Closed-form expression O(1) Closed-form expression O(1) Exact algorithm O(S)

Laurent DAUDET PhD Defense December 22nd, 2017 37 / 40

Minimizing the waiting time for a one-way shuttle service Results

Table of results

Pmax Pave Pmax

return

Pave

return

Exact algorithm O(K 2S) Exact algorithm O(K 2S)

? ?

Approx. algorithm O

• S log S

ε

• Approx.

algorithm O

SM 3

Approx. algorithm O

• β3SM 2S+1

?

Closed-form expression O(1) Closed-form expression O(1) Closed-form expression O(1) Exact algorithm O(S)

Laurent DAUDET PhD Defense December 22nd, 2017 37 / 40

Minimizing the waiting time for a one-way shuttle service Results

A conjecture for Pave

return

Conjecture There exists an optimal solution of Pave

return.

→ If such conjecture true, similar theorem than for Pmax

return. Laurent DAUDET PhD Defense December 22nd, 2017 38 / 40

Minimizing the waiting time for a one-way shuttle service Results

A conjecture for Pave

return

Conjecture There exists an optimal solution of Pave

return.

→ If such conjecture true, similar theorem than for Pmax

return. Laurent DAUDET PhD Defense December 22nd, 2017 38 / 40

Conclusion

General conclusion

Three main problems:

→ Operational scheduling problem with maximum number of shuttles. → Prospective scheduling and pricing problem with maximum revenue. → Theoretical scheduling problem with minimum waiting time.

Various methods:

→ Mixed Integer Linear Programming. → Stochastic Optimization and Sample Average Approximation. → Lagrangian relaxation. → Heuristics. → Exact algorithms and approximation schemes.

Laurent DAUDET PhD Defense December 22nd, 2017 39 / 40

Thank you for your attention.

Laurent DAUDET PhD Defense December 22nd, 2017 40 / 40